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ISSN: 2090-4908
International Journal of Swarm Intelligence and Evolutionary Computation
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Hybridization of Fruit Fly Optimization Algorithm and Firefly Algorithm for Solving Nonlinear Programming Problems

Rizk M Rizk Allah*

Department of Basic Engineering Science, El-Menoufia University, Shebin El-Kom, Egypt

*Corresponding Author:
Rizk M Rizk Allah
Department of Basic Engineering Science
El-Menoufia University, Shebin El-Kom, Egypt
E-mail: rizk_masoud @sh-eng.menofia.edu.eg

Received March 21, 2016; Accepted April 29, 2016; Published April 30, 2016

Citation: Allah RMR (2016) Hybridization of Fruit Fly Optimization Algorithm and Firefly Algorithm for Solving Nonlinear Programming Problems. Int J Swarm Intel Evol Comput 5: 134. doi: 10.4172/2090-4908.1000134

Copyright: © 2016 Allah RMR. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

We propose a novel hybrid algorithm named, FOA-FA to solve the nonlinear programming problems (NLPPs). The main feature of the hybrid algorithm is to integrate the strength of fruit fly optimization algorithm (FOA) in handling continuous optimization and the merit of firefly algorithm (FA) in achieving robust exploration. The methodology of the proposed algorithm consists of two phases. The first one employs a variation on original FOA employing a new adaptive radius mechanism (ARM) for exploring the whole scope around the fruit flies locations to overcome the drawbacks of original FOA which has been continues for the nonnegative orthant problems. The second one incorporates FA to update the previous best locations of fruit flies to avoid the premature convergence. The hybrid algorithm speeds up the convergence and improves the algorithm’s performance. The proposed FOA-FA algorithm is tested on several benchmark problems and two engineering applications. The numerical comparisons have demonstrated its effectiveness and efficiency.

Keywords

Firefly algorithm; Fruit fly optimization algorithm; Nonlinear programming problems

Introduction

Traditional optimization methods can be classified into two distinct groups; direct and gradient-based methods. In direct search methods, only objective function and constraint value are used to guide the search, whereas gradient-based methods use the first and/or second-order derivatives of the objective function and/or constraints to guide the search process. Since derivative information is not used, the direct search methods are usually slow, requiring many function evaluations for convergence. For the same reason, they can be applied to many problems without a major change of the algorithm. On the other hand, gradient based methods quickly converge to an optimal solution, but are not efficient in non-differentiable or discontinuous problems. In addition, if there is more than one local optimum in the problem, the result may depend on the selection of an initial point, and the obtained optimal solution may not necessarily be the global optimum. Furthermore, the gradient search may become difficult and unstable when the objective function and constraints have multiple or sharp peaks.

Therefore, to overcome these shortcomings, a lot of research has focused on meta-heuristic methods. The meta-heuristic methods have many advantages compared to the traditional nonlinear programming techniques, among which the following three are the most important: (i) They can be applied to problems that consist of discontinuous, nondifferentiable and non-convex objective functions and/or constraints. (ii) They do not require the computation of the gradients of the cost function and the constraints. (iii) They can easily escape from local optima. Many meta-heuristic algorithms such as, genetic algorithm (GA) [1], particle swarm optimization (PSO) [2], differential evolution [3] and ant colony optimization [4] have shown their efficacy in solving computationally intensive problems.

Among the existing meta-heuristic algorithms, a well-known branch is the FA which is a meta-heuristic search algorithm, has been developed by Yang [5,6]. FA mimics some characteristics of tropic firefly swarms and their flashing behavior. A firefly tends to be attracted towards other fireflies with higher flash intensity. The main advantages of FA are namely intensification and diversification or exploitation and exploration: As light intensity decreases with distance, the attraction among fireflies can be local or global, depending on the absorbing coefficient, and thus all local modes as well as global modes will be visited. Therefore, it has captured much attention and has been successfully applied to solve several optimization problems including [7-9].

Fruit fly optimization algorithm (FOA) is one of the latest metaheuristic methods which proposed by Pan [10] for solving optimization problems. The main inspiration of FOA is that the fruit fly itself is superior to other species in sensing and perception, especially in osphresis and vision. They feed chiefly on rotten fruits. In the process of finding food, their osphresis organs smell all kinds of scents in the air. They then fly towards the corresponding locations. When they get close to the food locations, they find foods using their visions and then fly towards that direction. FOA has many advantages such as a simple structure, ease of implementation and speed to acquire solutions. Therefore, FOA has been successfully applied to solve several optimization problems [10-14].

Recently, hybridization is recognized to be essential aspect of high performing algorithms. Pure algorithms cannot reach to an optimal solution in a reasonable time. In addition, the premature convergence in pure algorithms is inevitable. Thus pure algorithms are almost always inferior to hybridizations. To our best knowledge, this is the first time that the hybridization between FOA and FA has been proposed for solving NLPPs. This hybridization aims to improve the performance of the original FOA and eliminate its drawbacks.

In this paper, we propose a novel hybrid algorithm named, FOA-FA for solving the nonlinear programming problems and two engineering applications. The proposed FOA-FA algorithm employed the merits of both fruit fly optimization algorithm (FOA) and firefly algorithm (FA) and its methodology consists of two phases. The first one employs a variation on original FOA employing a new adaptive radius mechanism for exploring the whole scope around the fruit flies locations to overcome the drawbacks of original FOA which has been continues for the nonnegative orthant problems. In addition, several other improvements for original FOA are considered, such as: Initialization of the candidate solutions through the search space uniformly and a new modulation coefficient. Moreover, the premature convergence of original FOA degrades its performance by reducing its search capability, leading to a higher probability of being trapped to a local optimum. Therefore, the second phase incorporates the FA to update the previous best locations of fruit flies to force FOA jump out of premature convergence, because of its strong searching ability. Consequently, the hybrid algorithm speeds up the convergence and improves the algorithm’s performance.

The rest of the paper is organized as follows. In Section 2 we describe some preliminaries of the nonlinear programming problem. In Section 3, we introduce briefly the basics of both FOA and FA algorithms. The hybrid algorithm named, FOA-FA algorithm is proposed and explained in details in Section 4. The numerical experiments are given in Section 5 to substantiate the superiority of the proposed algorithm. Finally we summarize the paper with some comments.

Nonlinear Programming Problem (NLPP)

The nonlinear programming problem can be defined as follow [15]:

image                (1)

Where F(x) is the objective function defined on the search space S, imageX is called a decision variable and gk(x), hj(x) are defined the inequality and equality constraints, respectively. The sets imageand imagedefine the feasible and infeasible search spaces, respectively. Usually, the search space S is defined as an n-dimensional rectangle in image as in (2) (domains of variables defined by their lower and upper bounds):

image                (2)

Any point image is called a feasible solution, otherwise, X is an infeasible solution.

Overview of the FOA and the FA Algorithms

Fruit fly optimization algorithm (FOA)

In this section we describe the basic steps of the original Fruit fly optimization algorithm (FOA) in details as follows [10].

Step 1: Set the main parameters of FOA and give initial location for the fruit fly swarm randomly P

x-axis, y-axis

Step 2: Endow personal fruit fly with random direction for finding food by using:

xi = X- axis + Random value                (3)

Yi = y- axis + Random Value

i =1,2,…,m

where m is the swarm size of fruit flies

Step 3: Since the food location cannot be known, calculate the distance (Disti) of the fruit fly from the origin and find out the smell concentration judgment value (Si). Suppose that Si is the reciprocal (Disti) of as follows:

image

Step 4: Calculate the smell concentration (Smelli) of the individual fruit fly location by substituting the smell concentration judgment value (Si) into the smell concentration judgment function (also called Fitness function).

Smelli = Function (Si)         (6)

Step 5: Find out the individual fruit fly with the maximal smell concentration (the maximal value of Smell) among the fruit fly swarm:

[best smell best index] = max (Smell)                 (7)

Step 6: Keep the maximal smell concentration value and location (x, y) of the best fruit fly. Then, the swarm flies towards that location.

Smell best = Best smell

x-axis = x(best index)               (8)

y-axis = y(best index)

Enter iterative optimization to repeat the implementation of step 2–6. When the smell concentration is not superior to the previous iterative smell concentration any more, or the iterative number reaches the maximal iterative number, the circulation stops.

Firefly algorithm (FA)

The Firefly algorithm was developed by Xin-She [6] and it is based on idealized behavior of the flashing characteristics of fireflies. For simplicity, we can summarize these flashing characteristics as the following three rules:

a) All fireflies are unisex, so that one firefly is attracted to other fireflies regardless of their sex.

b) Attractiveness is proportional to their brightness, thus for any two flashing fireflies, the less bright one will move towards the brighter one. The attractiveness is proportional to the brightness and they both decrease as their distance increases. If no one is brighter than a particular firefly, it will move randomly.

c) The brightness of a firefly is affected or determined by the landscape of the objective function to be optimized [6].

For simplicity we can assume that the attractiveness of a firefly is determined by its brightness or light intensity which in turn is associated with the encoded objective function. In the simplest case for an optimization problem, the brightness I of a firefly at a particular position X can be chosen as I (x) ∝1 f (x). However the attractiveness is relative, it should vary with the distance rijbetween firefly i and firefly j. As light intensity decreases with the distance from its source and light is also absorbed in the media, so we should allow the attractiveness to vary with degree of absorption.

Step 1: Attractiveness and light intensity

In the firefly algorithm, there are two important issues: the variation of the light intensity and the formulation of the attractiveness. We know, the light intensity I(r) varies with distance r monotonically and exponentially, that is:

image                (9)

Where I0 the original light intensity and γ is the light absorption coefficient. As firefly attractiveness is proportional to the light intensity seen by adjacent fireflies, we can now define the attractiveness β of a firefly by Equation (10)

image                (10)

Where r is the distance between each two fireflies and β0 is their attractiveness at r = 0 i.e., when two fireflies are found at the same point of search space. The value of γ determines the variation of attractiveness with increasing distance from communicated firefly.

Step 2: Distance

The distance between any two fireflies i and j at xi and xj respectively, the Cartesian distance is determined by equation (11) where xi,k is the kth component of the spatial coordinate of xi the ith firefly and n is the number of dimensions.

image                (11)

Step 3: Movement

The firefly i movement is attracted to another more attractive (brighter) firefly j is determined by:

image                (12)

Where the second term is due to the attraction while the third term is randomization with α being the randomization parameter and r and is a random number generator uniformly distributed in [0, 1].

The Hybrid FOA-FA Algorithm

In this section, the proposed algorithm will be presented in detail. The main reason for developing the FOA-FA is to overcome the drawbacks of the original FOA which cannot handle the negative domain that is the candidate solution cannot be uniformly generated in the problem domain. Also original FOA is easy to fall into premature convergence because the random term of the Equation (3) picks small values within a radius equal to one around the best location.

The methodology of the proposed FOA-FA algorithm consists of two phases. The first one employs the FOA, where a swarm of fruit flies that moves in various directions using the ARM. Therefore, these moves will follow a uniform distribution. The other phase incorporates the FA to update the previous best locations of fruit flies to force FOA jump out of premature convergence, because of its exploitation and exploration ability. Consequently, the hybrid algorithm speeds up the convergence and improves the algorithm’s performance. The main steps of the proposed algorithm as follows:

Step 1: Initialize the parameters

The main parameters of the proposed algorithm are the maximum iterationnumber (T),thewarmsize(m) and distance zone of fruit fly (R).

The main parameters of the proposed algorithm are the maximum iteration number (T),thewarmsize(m) and distance zone of fruit fly (R).

Step 2: Initialize the swarm location

Initial fruit fly swarm location is accompanied within the search space, that is, the algorithm assigns a random vector image for i-th fruit fly as in Equation (13):

image                (13)

Where rand is a random value from the uniform distribution on the interval [0, 1].

Step 3: Solution construction using osphresis

The osphresis that introduced in the original FOA (see Equation 3) was picked small values within a radius that equals one. So, we introduce the ARM for enhancing the diversity of solutions and achieving an effective exploration to the whole scope around the fruit flies locations, where the search radius decreases dynamically with iteration number as shown in Figure 1 (i.e., R=100, T=100). Therefore, the process of the solution construction using osphresis is implemented as follows:

Figure

Figure 1: The variation of Δi(t,R) versus the number of iterations.

image                (14)

Where R is the search radius that determines the extent of the search subspace which we want to cover during the run and ε is a random number from [0, 1]. The random step Δ(t,R) returns a value in the range [0, R] such that the probability of Δ(t,R) being close to 0 increases as t increases. The solution construction using osphresis is demonstrated in Algorithm.

Algorithm 1. The solution construction using osphresis

1. Input : image

2. set image

3. image

4. if imageset image then image

5. elseif imageset image then image

6. end if

7. t = t +1

8. i = i +1

9. Output: xi ∈Ω

Step 4: Evaluation

The smell concentration ( Smelli ) of the individual fruit fly is related to objective function (fitness function). In this step, the fitness function is evaluated based on the decision variable X as in Equation (15) instead of using the smell concentration judgment value (Si). Therefore, the best food source with the lowest fitness xbest is also found as in Equation (16), i.e., image.

Smelli = f(xi), i= 1,2,..m           (15)

image                (16)

Step 5: Update the swarm location

In this step, the swarm flies toward the best obtained location using Equation (16). If Xbest is better than the current fruit fly swarm location Xi- axis, it will replace the swarm location and become a new one in the next iteration, i.e., image.

Step 6: Crossover operator for updating the osphresis

In this step, the crossover operator is adopted for enhancing the osphresis between fruit flies as follows:

image                (18)

where image is the coefficient of modulation, it decides the scope of modulation, image is random numberand isthe extending-modulation operator is specified by the user. The idea of this step is to make the new generation of fruit flies around the best fruit fly, where the new solution not within xi(t) and xbest(t) but the segment may be extended equally on both sides (i.e., 3γ-3 as in Figure 2.

Figure

Figure 2: The demonstration of updating the osphresis.

Therefore, the best food source with the lowest fitness xbest is also found as in Equation (16), i.e.,image. If xbest is better than the current fruit fly swarm location Xi-axis , it will replace the swarm location and become a new one in the next iteration, i.e., image.

Step 7: FA phase for updating the diversity

The random term of the Equation (3) makes the fruit flies swarm fluctuates around the so far best solution; consequently, the diversity loss occurs. Under this condition, the best solution might be easily trapped in a local optimum or premature convergence. FA is incorporated to update the previous best locations of fruit flies to force FOA jump out of premature convergence, because of its exploitation and exploration ability. Consequently, the hybrid algorithm speeds up the convergence and improves the algorithm’s performance. In the rest of this paper, we will discuss the implementation FA in three steps as follows (Figure 3).

Figure

Figure 3: The pseudo code of the proposed FOA-FA algorithm.

Step 7.1: Initialization: Initialize a swarm of fireflies with the obtained best ant position, where each firefly contains n variables (i.e., the position of the ith firefly in the n dimensional search space can be represented as image). Furthermore every member of the swarm is characterized by its light intensity (i.e., initialize each firefly with distinctive light intensity. Ii(xi), i = 1, 2, …m

Step 7.2: Light intensity

Calculate the light intensity Ii(xi), i = 1, 2, …m for each firefly which in turn is associated with the encoded objective function, where for minimization problem the brighter firefly represents the minimum value for I(x).

Step 7.3: Attractiveness and movement

In the firefly algorithm, the attractiveness of a firefly is determined by its light intensity which in turn is associated with the encoded objective function and the distance rij between firefly i and firefly j. In addition, light intensity decreases with the distance from its source, and light is also absorbed in the media, so we should allow the attractiveness to vary with the degree of absorption. As a firefly’s attractiveness is proportional to the light intensity seen by adjacent fireflies, we can now define the attractiveness β of a firefly by Equation (10). On the other hand, the movement of a firefly i is attracted to another more attractive (brighter) firefly J is determined by Equation (13).

image                (13)

Since each member of the swarm explores the problem space taking into account results obtained by others, therefore the randomization term may moves the fire fly to lose its best location, so we introduce a modification on the randomization term that makes the fireflies approached from the optimum. A further improvement on the convergence of the algorithm is to vary the randomization parameter α1 so that it decreases gradually as the optima are approaching. For example, we can use

image                (14)

Where Tc is the maximum number of generations for FA, and θ є [0, 1] is the randomness reduction constant.

The basic steps of the proposed FOA-FA algorithm can be summarized as the pseudo code shown in Figure 4.

Figure

Figure 4: The graphical representation for the evaluation of metric.

Results and Discussions

In this section, performance of the proposed FOA-FA algorithm is tested on eleven benchmark problems, which are known from the literature and reported in Young [16] and two engineering applications [17,18]. The solutions of the proposed FOA-FA are compared with the prominent algorithms that reported in Young [16-18]. The test problems are listed in Table 1. The algorithm is coded in MATLAB 7, which has an Intel Core I 5 (1.8 GHz) processor and 4 GB RAM memory.

Figure

Table 1: Test problems.

Table 1: Test problems.

On the other hand, extensive experimental tests were conducted to see the effect of different values on the performance of the proposed algorithm. Based upon these observations, the following parameters have been set as in Table 2.

Number of iteration (T) 200
The swarm size (m) 100
Initial attractiveness (β0) 1
The light absorption coefficient(γ) 1
The randomness reduction constant(Ѳ) 0.96
Randomization parameter (α) 0.95

Table 2: Parameters settings.

The performance evaluation metrics

Our hybrid algorithm was tested on set of the test problems in engineering domain. Each of the test problems was run thirty times independently, with different seeds. In order to evaluate the closeness between the obtained optimal solution and the theoretical optimal solution, as well as measuring the number of evaluations for the proposed algorithm and the existing algorithms, we used two different metrics.

The speed: The speed with which the optimum is obtained is a criterion for examining the performance. For the proposed FOA-FA to be competitive, it is important to be able to compare its speed with the other existing algorithms. As time is a relative notion depending on the computer used, we preferred to measure the speed by counting the number of evaluations of the tested problem. On the one hand, this criterion is independent from the type of computer used, and on the other hand the evaluation of the function is the operation taking the longest time in the program.

The error: So the test for closeness, we used the evaluation of the error between the result given by the algorithm and the exact value of the optimum, for each test problem. The relative error was used each time it was possible:

image               (20)

When the optimum to reach was 0, it was no longer possible to use this expression so we calculated the absolute error:

image               (21)

Table 3 compares the performance of the proposed FOA-FA, where the better values in the comparison are bold in font type. On the other hand, a comparison is made with the prominent algorithms from the literature. The test problems have been solved by FOA-FA for 30 times. The starting values of the variables for each problem were selected randomly for all runs from the solution space. The results found by FOA-FA such as the best and average function value, numbers offunction evaluation (NFE) and solution time in seconds have been recorded in Table 3, whereas for the other algorithms only the function value and numbers of function evaluation are given because the solution times, the best and average function value for some algorithms not given. First, we start by comparing the optimum solution of the given the test problems for the proposed FOA-FA algorithm and the existing algorithms. Based on the obtained results that demonstrated in Table 3 we notes that, the proposed FOA-FA is successful finding the optimum solution for the test problems F2 and F4 better than both the theoretical optimum solution and the obtained by other existing algorithms. Also the proposed FOA-FA algorithm outperforms the existing algorithms for functions F1, F6, F7, F8, F9, F10 and F11 but equals the SZGA for the test problems F6, F7 and F8. Second, as shown from Table 4, the proposed algorithm speeds up the convergence through inclusion of the lowest evaluations, where the proposed algorithm elapsed number of evaluations less than the other existing algorithms for finding the global optimum as in Figure 4, that is the proposed algorithm is the faster than the existing algorithms.

 F (x)  FOA-FA  Compared algorithm
 Function value  Number of function evaluation(NFE) Time(s) Name Function value NFE
Best Average Best Average Best Average  
F1 0 1.792475E-20 1500 1500 1500 0.629180 SGA
MGA
SZGA
0.44108E-5
0.40842E-5
0.29802E-7
4000
18182
4000
F2 - 16.09172000 6 -16.09172 2000 2000 2000 0.323088 SGA
MGA
SZGA
-16.09171
-16.09171
-16.09172
4000
8000
4000
F3 0.9980 0.9980 1500 1700 1500 0.766544 SGA
MGA
SZGA
0.9980
0.9980
0.9980
8000
2000
2000
F4 0.39785 0.3978873 2500 2500 2500 0.401413 SGA
MGA
SZGA
0.39789
0.39789
0.3979
8000
10000
4000
F5 -1.03163 -1.0316284 1600 2000 1600 0.134047 SGA
MGA
SZGA
-1.03163
-1.03163
-1.03163
6000
3000
3000
F6 3 3 2500 2500 2500 0.716377 SGA
MGA
SZGA
3.0001
3.0001
3
6000
5000
4000
F7 -186.73091 -186.730908 2300 2500 2300 4.8243 SGA
MGA
SZGA
-186.72744
-186.72825
-186.73091
6000
12000
3000
F8 8.01276 8.0137 10000 10000 10000 19.6132 SGA
MGA
SZGA
8.01359
8.01293
8.01276
200000
270270
20000
F9 3.6284E-7 3.6284E-7 60000 60000 60000 18.6979 SGA
MGA
SZGA
1.63594
0.44407E-2
0.13074E-5
200000
317460
175440
F10 0 7.5684E-17 25000 24500 25000 4.799961 SGA
MGA
SZGA
0.15029E-1
0.13565E-1
0.25422E-7
400000
320000
320000
F11 1.3022E-15 9.4984E-7 2500 2500 2500 15.0280 SGA
MGA
SZGA
0.89927
0.16851
0.23033E-3
400000 49855
70000

Table 3: The comparison of solution quality.

Problems Absolute error Relative error
Proposed algorithm Compared algorithms Proposed algorithm Compared algorithms
F1 0 SGA
MGA
SZGA
0.44108E-5
0.40842E-5
0.00298E-5
- SGA MGA
SZGA
-
-
-
F2 0.6E-8 SGA
MGA
SZGA
0.99999E-5
0.99999E-5
0
-0.37286E-9 SGA
MGA
SZGA
-0.62143E-6
-0.62143E-6
0
F3 0 SGA
MGA
SZGA
0
0
0
0 SGA
MGA
SZGA
0
0
0
F4 0.4E-4 SGA
MGA
SZGA
0
0
0.99999E-5
E-30.10053 SGA
MGA
SZGA
0
0
0.251325E-4
F5 0 SGA
MGA
SZGA
0
0
0
0 SGA
MGA
SZGA
0
0
0
F6 0 SGA
MGA
SZGA
0.1E-3
0.1E-3
0
0 SGA
MGA
SZGA
0.3333E-4
0.3333E-4
0
F7 0 SGA
MGA
SZGA
0.003469
0.002659
0
0 SGA
MGA
SZGA
-0.18582E-4
-0.14245E-4
0
F8 0 SGA
MGA
SZGA
0.83E-3
0.17E-3
0
0 SGA
MGA
SZGA
0.10358E-3
0.021216E-3
0
F9 0.36284E-6 SGA
MGA
SZGA
1.63594
0.44407E-2
0.13074E-5
- SGA
MGA
SZGA
-
-
-
F10 0 SGA
MGA
SZGA
0.15029E-1
0.13565E-1
0.25422E-7
- SGA
MGA
SZGA
-
-
-
F11 0.13022E-14 SGA
MGA
SZGA
0.89927
0.16851
0.23033E-3
- SGA
MGA
-
-
-

Table 4: The performance assessment.

Finally, the last performance metric is the relative error or the absolute error, where as depicted from Table 4, the absolute error by using the proposed algorithm is better than the other exists algorithms for all test problems except SZGA for the test problem F4 and also the relative error is superior to the other exists algorithms for all test problems except SZGA for the test problem F4 The SZGA seems to better than the proposed algorithm for the test problem F4 this because the error is calculated related to the theoretical optimum. In fact, the proposed algorithm outperforms the theoretical optimum for the test problem F4, where if the relative error and the absolute error are calculated related to the obtained optimum by our algorithm we find it 0.12567 E-5 and 0.5 E-6 respectively. Therefore, the simulation results prove superiority of the proposed algorithm to those reported in the literature, where it is completely better than the other algorithm

Design optimization problems

The design optimization problems are a very important research area in engineering studies because real world design problems require the optimization of relevant engineering applications. In terms of robustness and efficiency of the available methods, these methods are still in need of improvements. Therefore, to meet the ever increasing demands in the design of the electrical devices, the proposed FOA-FA has been substantiated its capability for solving these applications. Thus, the validity of the proposed algorithm has been proved superiority by using two engineering applications, the linear synchronous motor (LSM) and air-cored solenoid [17,18].

Design of LSM: The linear synchronous motor (LSM) operates on the same working principle as that of a permanent magnet rotary D.C. motor [17]. As in a rotary motor there are two parts in a LSM, one is the set of permanent magnets and the other is the armature that has conductors carrying current. The permanent magnets produce a magnetic flux perpendicular to the direction of motion. The flow of current is in the direction perpendicular to both the direction of the motion and the direction of the magnetic flux.

To validate the proposed algorithm, it is employed to optimize the geometrical design of the linear electric actuator problem as is shown in Figure 5(a) and 5(b). The objective function is to maximum force subject to some of constraints on heat, radius, saturation, demagnetization and maximum force constraint. There are four design variables: the current in each slot, x1, the dimensions of the slot, and x2 x3, and the height of the magnet, x4. The mathematical formulation of the objective function is described as follows:

Figure

Figure 5: The configurations of two designs.

image

The obtained result for design of LSM is demonstrated in Table 5, by giving the best, worst and mean values for the objective function. Also Figure 6 depicts the relation between the maximum force and number of iterations for FOA before applying FA and after Applying FA (i.e., the hybrid algorithm FOA-FA). The results of the proposed FOAFA is superior to those obtained using the Deshpande [17].

Methods FOA-FA FOA Deshpande [13]
Best objective function value 17705 1.6239E4 16192
Worst objective function value 12139 15440 -
Mean objective function value 1.7043E4 1.6198E4 -
Median objective function value 17703 1.6239E4 -
NFE 4000 4500 -
Time (second) 3.449 0.3812 -

Table 5: Optimal results of the LSM problem.

Figure

Figure 6: Maximum force by using FOA and FOA-FA.

Design of an air-cored solenoid: The optimization problem of a coreless solenoid with rectangular cross section x1×xi and a mean radius x3 Figure 5(b) is tackled from Barba [18]. This problem can be formulated as: maximize the inductance L(x1, x2, x3 ) and satisfy some of constraints for the given length k1 = 10 m and k2 = 10-6 of the current carrying wire. In order to simplify the analysis, two variables, x1 and x2, are considered. Correspondingly, the optimization problem is simplified as:

image

Table 6 demonstrates the result for design of air-cored solenoid, by giving the best, worst and mean values for the objective function. On the other hand, Figure 7 depicts the relation between the maximum inductance against the number of iterations for FOA before applying FA and after Applying FA (i.e., the hybrid FOA-FA). The result of the proposed FOA-FA is superior to those obtained by [18].

Methods FOA-FA FOA Barba[14]
Best objective function value 258.0607 258.0291 257.197
Worst objective function value 250.0581 244.3131 -
Mean objective function value 257.7696 257.5935 -
Median objective function value 257.9143 258.0084 -
NFE 3300 4477 -
Time (second) 1.2345 0.4378 -

Table 6: Optimal results of the air-cored solenoid.

Figure

Figure 7: Maximum inductance by using FOA and FOA-FA.

Figure 8 shows the graphical representation of the optimum solution for the LSM and air cored solenoid. It is obvious that the algorithm outperforms the other algorithms for the both problems.

Figure

Figure 8: The graphical representation of the optimum solution.

We concluded that the integrated algorithm of FOA with FA has improved the quality of the founded solutions and also it guarantees the faster converge to the optimal solution. The FOA is presented in the first stage to provide the initial solution to optimization problem (close to the optimal solution as possible) followed by FA to enhance the quality of the solution. However, because of its random behavior, FOA may suffer from slow convergence. So we introduced the neighborhood search via adaptive radius mechanism Δ (t,R) that returns a value in the range [0, R]. On the other hand, to reach a quick and closer result to optimal solution, and to improve the efficiency of the FOA, hybridization between FOA and FA was implemented.

In this subsection, a comparative study has been carried out to assess the performance of the proposed FOA-FA algorithm concerning the hybridization, closeness to optimal solution and computational time. On the first hand, pure algorithms suffer from reach to an optimal solution in a reasonable time. Also, the sinking into premature convergence is inevitable in pure algorithms. Consequently, our hybridization algorithm has twofold feature, solving NLPPs in a reasonable time and avoiding the sinking into the premature convergence. On the other hand, the proposed FOA-FA algorithm is highly competitive when compared with the other existing methods in term of calculating the relative, absolute error and number of evolutions. Accordingly, it provides the facility to search in the negative and the nonnegative orthant through adaptive radius mechanism. Finally, the feasibility of using the proposed algorithm to handle NLPPs has been empirically approved.

Conclusions

A novel hybrid algorithm named, FOA-FA for solving the nonlinear programming problems and engineering applications is proposed. The proposed FOA-FA algorithm employed the merits of both fruit fly optimization algorithm (FOA) and firefly algorithm (FA) and consists of two phases. The first one employs a variation on original FOA employing a new adaptive radius mechanism for exploring the whole scope around the fruit flies locations to overcome the drawbacks of original FOA, which has been an continues to be a fruitful paradigm for designing effective the nonnegative orthant algorithms. Moreover, the premature convergence of original FOA degrades its performance by reducing its search capability, leading to a higher probability of being trapped to a local optimum. Therefore, the second phase incorporates the FA to update the previous best locations of fruit flies to force FOA jump out of premature convergence, because of its strong searching ability. The hybrid algorithm speeds up the convergence and improves the algorithm’s performance.

A careful observation will reveal the following benefits of the proposed optimization algorithm.

1) It can efficiently safeguard the solution from sinking into the premature convergence through incorporating the FA.

2) It can overcome the lack of the exploration of the FOA which roaming in the nonnegative orthant by introducing the adaptive radius mechanism.

3) It emphasizes the diversity of solutions by incorporating the FA that enriches the exploratory capabilities of the proposed algorithm.

4) It competitive when compared with the other existing algorithm.

5) It can find can find the global minimum for the problems very efficiently.

6) It can accelerate the convergence and improves the algorithm’s performance through elapsed low computational time.

The future work will be focused on three directions: (i) the application of FOA- FA to real world problems; (ii) the extension of the method to solve the multi-objective problems; and (iii) another applications in engineering.

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