Fitting nonlinear mathematical models to the cost function of the quadrafilar helix antenna optimization problem

Abstract

Here, optimization of a quadrafilar helical antenna is presented to compare the performances of objective function pairs adapted from mathematical models by using DEA with fixed weight objective function structure and SPEA2 with variable weight objective function structure from 2 different competitive multi-objective algorithms. The most important purpose in optimization problems is to find the result with the lowest cost. For this, the selection of the appropriate objective function pair is very important. The most important aim in this study is to determine the optimum objective function pair model. For this purpose, five different objective function models were derived by using nonlinear mathematical models. These objective functions are adapted from polynomial, power, exponential, gaussian and fourier mathematical models. In order to determine the most successful model without question, the objective functions adapted from the mathematical models are compared separately in both evolutionary algorithms by using different algorithm parameters and different weight coefficients. According to the results obtained, it is seen that the objective function adapted from the power mathematical model has the lowest cost. This proposed adaptation technique, which is the novelty of the study, is an efficient and reliable method to find the most appropriate objective function and the lowest cost result in optimization problems. It can also be quickly adapted to any optimization problem.

Introduction

Optimization is called a method that reaches the determined goals by using the available resources in the system in the most appropriate way [1]. Optimization consists of two basic components; these are called modeling and solving. The first component, namely modeling, is the mathematical expression of any problem. The second component analysis, on the other hand, is to obtain the optimum solution that satisfies the modeled expression. With the advent of optimization methods, researchers first took great interest in modeling. Although modeling is practical and faster, they have turned to the analysis component because of its limited boundaries. Since the optimization methods are economical and practical, and they are needed in many problems, the development of the analysis component continues at great speed. Problems encountered in real life have more than one purpose. Therefore, the need for multi-objective optimization problems is increasing. In order to overcome multi-objective problems, researchers have developed new methods that can find effective results covering the entire solution space and combined these methods with optimization algorithm models. These multiple purposes are combined into a single purpose or expressed separately. These methods are named as fixed-weighted objective function and variable-weighted objective function, respectively [2, 3]. It may not always be possible to collect these criteria in a single objective function. In problems where more than one criterion is in question, especially in which these criteria conflict with each other, different solution alternatives are in question. Multi-objective optimization problems have more than one solution. Using the algorithms used in the solution of single-objective problems in the solution of such problems can sometimes lead to results such as not scanning the solution space enough and not getting good results. Solving single-objective optimization problems is of course easier than solving multi-objective optimization problems. The existence of conflicting objectives in multi-objective optimization problems increases the difficulty of such problems. For example, trying to maximize one of the goals while trying to minimize the other goal further increases the complexity of the problem [4]. In short, it is concluded that the determination of the optimum objective function in optimization problems comes before everything else.

The Quadrafilar helical antenna (QHA), invented by Gerst, has been extensively studied over the past few decades [5, 6, 7–8]. The QHA consists of four equally spaced helices on a cylinder, and each helix is connected to an equal amplitude feed network with relative phases of 0°, 90°, 180° and 270°. This antenna is suitable for mobile satellite radio receiver antenna due to its large CP beam width and low back radiation regardless of the reduced ground plane size. Recently, QHA has been shrinking in size for use in handheld mobile devices. The smaller the size of the QHA, the greater the mutual coupling between each helical antenna. There are many studies on HA and QHA optimization and design in the literature [5, 6, 7, 8, 9–10]. Most of these studies have a single objective function pair or single decision variable. There is a limited study in the literature, which has been applied to the modeling component of optimization, which is about adapting mathematical models to the cost function [11]. However, there are different studies combining mathematical modeling and optimization [12, 13, 14–15]. However, such a large study applied to the analysis component of optimization has not yet been encountered. When all these studies are examined, the idea arises that a successful study can emerge by combining these subjects. Both the optimization of more than one decision variable in the antenna design and the quadrafilar nature of the antenna puts the optimization into a very difficult process. This challenging problem will be overcome by using original objective functions not yet encountered in the literature. This study is an original study in terms of adapting the objective functions of the analysis component in optimization with mathematical models and having a double decision variable. In addition, in any optimization work to be done, it will make a great contribution to the users in the selection of the objective function pair.

The remainder of this article is organized as follows: Sect. 2 provides information about the antenna to be used. The optimization algorithms and design parameters used are mentioned in Sect. 3. The main idea of the study, objective and cost functions are discussed in Sect. 4. The results analyze of the study were done in Sect. 5. After the discussion and future work is covered in Sect. 6, the article ends in Sect. 7.

Quadrafilar helix antenna

A helical antenna consists of one or more circular or differently wound conductive wires. The most common type, helical antenna monofilar made of a single coiled wire, is called bifilar and quadrafilar, consisting of two or four wires, respectively. In most cases, directional helical antennas may not be omnidirectional designs in a ground plane. The supply line is connected between the base of the helix and the ground plane. The polarization of the antenna can be determined by the value defined as the axial ratio. If the antenna has two separate components in the Θ and Φ directions, and $\left|E_{\mathrm{\Theta }}\right|=\left|E_{\mathrm{\Phi }}\right|$ the antenna has mostly elliptical polarization. The absence of components in the Θ or Φ direction of the antenna ensures that the antenna has vertical or horizontal linear polarization. The closer this ratio is to 1, which is calculated separately for each frequency and angle, the better circular polarization the antenna will have. The importance of circular polarization is that it can receive signals from antennas with horizontal and vertical polarization, so the axial ratio is of great importance when designing antennas. In the normal mode helical antenna, the axial ratio $\mathrm{\Theta }=\mathit{\pi }/2$ is calculated [16]. One of the most important advantages of the axial mode helical antenna is that it has circular polarization in the broadband. In this way, it can receive signals from antennas with horizontal and vertical polarization. The axial ratio, which determines the polarization and takes values from 0 to, is close to 1 in the wide frequency band for this antenna type. In the literature, the antenna has circular polarization at frequencies where the axial ratio is less than 3 dB. The axial ratio also improves as the number of turns increases, becoming less than 3 dB in a larger portion of the band. Although increasing the number of turns improves the axial ratio and enables the antenna to make circular polarization in a wider band, it is disadvantageous that it increases the physical length of the antenna too much. While the increase in diameter improves the axial ratio at the beginning of the band, it causes a deterioration at the end of the band [16]. Circular helical antennas are frequently encountered in wireless communication systems [16, 17]. There are also helical antennas in different design types [18, 19, 20, 21, 22, 23, 24, 25, 26–27]. In this study, a QHA model consisting of four wires will be used. Although there are many empirical equations [28, 29] for helical antenna design, MATLAB 2021's antenna toolbox is used in optimization processes and antenna simulations because these equations are inconsistent and not suitable for practical designs [30, 31]. Here, the QHA given in Fig. 1 is taken into account for the implementation of Differential evolution algorithm (DEA) and Strength pareto evolutionary algorithm (SPEA)-2.

[See PDF for image]

Fig. 1

Quadrafilar helix antenna

Optimization algorithms and design parameters

DEA belongs to the class of evolutionary algorithms that use differences in solution vectors and create new solutions with them. These new solutions are based on population differences. The basis of this is the population orientation and distribution, as well as the differences in the members of the population. According to the central limit theorem, as the population size increases, the distribution behind the differential evolution algorithm sampling technique tends towards the multivariate gaussian (normal) distribution, which is very common in the context of evolutionary algorithms. DEA is a very simple, yet very powerful and useful algorithm and can be used to deal with a wide variety of optimization problems [32, 33, 34, 35, 36–37]. SPEA2 has been developed as an enhanced version of the SPEA. This developed algorithm has a mechanism such as sorting of population members and k-Nearest Neighbor (kNN) as well as mutation—crossover parameters used in genetic algorithms. SPEA2 is one of the most useful and famous multi-objective optimization algorithms. It is also widely used to solve real-world, scientific and engineering applications [38, 39, 40, 41, 42–43]. The systematic and simultaneous optimization of more than one objective in performance parameters is called multi-objective optimization. In such problems, it is a big problem to choose the objective functions while creating the decision variable model. Since the problem is multi-objective, decision variables contain multiple criteria to evaluate the performance of the solution. Optimization models consist of mathematical expressions that reflect the operation and characteristics of the system, and the interactions with other systems in and around the system. These are the parameters that determine the measurable features of the system, the variables that determine the decision values that will give the best results, the performance criteria that will optimize the system, and the constraints that determine the limits of the system. The parameters that determine the measurement functions of the system are given in Table 1. In addition, as a constraint of the system, the ground plane radius of minimum 35 (mm) was adjusted during coding. Four of the parameters that determine the performance of the system given in Table 1 have been chosen so that the others are constant. In this optimization process, the decision variables are radius of turns (mm), width of strip (mm), spacing between turns (mm), and ground plane radius (mm). The objective functions that will determine the best decision variables, which are the novelty of the study, will be explained in detail in the next section.

Table 1. Antenna design parameters

Parameter	Value	Definition
Radius	13–19.5 (mm)	Radius of turns
Width	0.58–0.87 (mm)	Width of strip
Spacing	18.5–27.7 (mm)	Spacing between turns
GroundPlaneRadius	35–66 (mm)	Ground plane radius
NumArms	4	Number of helical elements
Turns	3	Number of turns
WindingDirection	CW	Direction of helix turns (windings)
Conductor	PEC	Type of metal material
FeedStubHeight	1 (mm)	Feeding stub height from ground

Objective and cost functions

Multi-objective optimization problems have more than one solution. Using the algorithms used in the solution of single-objective problems in the solution of such problems can sometimes lead to results such as not scanning the solution space enough and not getting good results. In order to achieve effective results in the solutions of multi-objective problems, researchers have developed methods that cover the entire solution space and adapted them to solution algorithms. The fixed-weight objective function combines multiple objectives under a single objective and makes them a single objective problem [2, 44]. The variable-weighted objective function was created for the first time in a study to eliminate the deficiencies in the fixed-weighted objective function method [2, 3]. In the study, DEA and SPEA2 were preferred, respectively, since they are examples for both species. Optimization is mathematically finding the value that makes an objective function optimal within a given definition range. From this, it can be concluded that one of the most important points of optimization is the correct determination of the objective function. In order to create the objective functions in the optimization process, two reference points, S₁₁ from the measurement functions and 30-degree directivity gain, were determined as decision variables. In multi-objective optimization problems, it is tried to converge to zero by giving priority to 2 objective functions at the rate of determined weight coefficients. Here, it is aimed to have a high directivity value and to find S₁₁ as low as possible. Accordingly, 5 different objective functions will be adapted based on mathematical models.

Adapting mathematical models to objective functions

Five different nonlinear objective function pairs were determined for the optimization of the quadrafilar helical antenna. These he called the polynomial (1), power (2), exponential (3), gaussian (4) and fourier (5) models. The determination of all models is explained in detail. The basis of the selected mathematical models is available in a study [45].

1

$$z=a_0\ast x+a_1\ast y\dots$$

2

$$z=a_0\ast x^2+a_1\ast y^2\dots$$

3

$$z=a_0\ast e^{\mathit{ax}}+a_1\ast e^{\mathit{by}}\dots$$

4

$$z=a_0\ast e^{a{x}^2}+a_1\ast e^{b{y}^2}\dots$$

5

$$z=cos{a}_0\ast x+sin{a}_1\ast y\dots$$

Polynomial Model In mathematics, a polynomial is defined as an expression consisting of independent variables and a constant number. The expression includes operations such as addition, subtraction, multiplication, division and exponentiation. The coefficient in exponentiation determines the degree of the polynomial. Here, the most basic and simple form of first order, which includes only multiplication and division, is preferred. In this context, positive division is applied to the value that is desired to be kept high, and negative division is applied to the value that is desired to be kept low. In addition, the weight coefficients of the variables were added to the expression as a product. According to all these, the following objective functions are defined:

6

$$O{F}_{11}=min\left\{\frac{w{c}_1}{\mathit{directivity}}\right\}$$

7

$$O{F}_{12}=min\left\{\frac{w{c}_2}{-S11}\right\}$$

Power Model Power model fitting is the variable that determines which one will be stronger in the change between two functions. In fact, it can be assumed to be a second-order or higher-order version of the polynomial model described above. Here, the quadratic form where the exponent will be 2 is preferred for both objective functions. The weight coefficient value was also taken as the exponent and included in the power model. According to these definitions, the following objective functions are defined:

8

$$O{F}_{21}=min\left\{{\left(\frac{w{c}_1}{\mathit{directivity}}\right)}^2\right\}$$

9

$$O{F}_{22}=min\left\{{\left(\frac{w{c}_2}{-S11}\right)}^2\right\}$$

Exponential Model The exponential function, or exponential function, is one of the frequently used functions in mathematics. Its general definition is $a^x$. Here, e is often used instead of a. Here, the value to be kept high is written as negative exponent and the value to be kept low is written as positive exponent. As in all other models, the weight coefficient value is added to the expression as a multiplier. In this way, the following objective functions are defined:

10

$$O{F}_{31}=min\left\{w{c}_1\ast e^{-directivity}\right\}$$

11

$$O{F}_{32}=min\left\{w{c}_2\ast e^{S11}\right\}$$

Gaussian Model The Gaussian model simulation is actually done with an approach similar to the transition from the polynomial model to the power model. For the expression whose general definition is $a^x$, the exponent is taken as 2 within the x value. Unlike the power model, since the result of $e^x$ does not decrease or increase linearly, the weight coefficient value is excluded from the exponent. According to all these definitions, the following objective functions are defined:

12

$$O{F}_{41}=min\left\{w{c}_1\ast e^{-directivit{y}^2}\right\}$$

13

$$O{F}_{42}=min\left\{w{c}_2\ast e^{S{11}^2}\right\}$$

Fourier Model Based on the Fourier conjecture, it is possible to represent expressions with cosine and sine curves. In order to maximize the values positively or negatively, the cosine curve is preferred. Again, as in every function, weight coefficients are added as multipliers. Accordingly, the following objective functions are defined:

14

$$O{F}_{51}=min\left\{w{c}_1\ast cos\left(\mathit{directivity}\right)\right\}$$

15

$$O{F}_{52}=min\left\{w{c}_2\ast cos\left(S11\right)\right\}$$

Cost function

In standard optimization problems, the cost function is obtained by adding the 2 determined objective functions. According to this definition, the 1st cost function is defined as follows.

16

$$cos{t}_1=O{F}_{i1}+O{F}_{i2},i=1,2,\dots ,5$$

However, in this problem, an additional single cost function has to be determined in order to be able to compare the success of the objective functions equally. For this, the objective function pair in the polynomial model, which is the most basic, will be taken as a basis. By summing the objective functions, a common 2nd cost function is obtained as follows:

17

$$cos{t}_2=\frac{w{c}_1}{\mathit{directivity}}+\frac{w{c}_2}{-S11}$$

Since analysis is required for the decision variables specified beforehand at the 3.5 GHz frequency, the objective function pairs (6–7), (8–9), (10–11), (12–13) and (14–15) to be used in the optimization process were selected. The result was tried to be determined with the minimum cost_1–2 (16,17) taken over 10 times with the determined objective functions.

In the next section, a detailed working case on optimization of QHA with predetermined design parameters for the 3.5 GHz frequency band will be presented.

Results analyzes

In the study analysis part, firstly, the optimal parameters for both algorithms (DEA & SPEA2) will be determined separately. In the next section, optimization processes will be continued using these determined parameters and the performances of the objective function pairs adapted from the mathematical models will be compared among themselves. In the next section, tests will be made for different weight coefficients (wc₁, wc₂), since the requirements of the measurement functions in the decision variable may be different. At the end of the chapter, measurement functions selected as decision variables for some of the results will be obtained as 3D electromagnetic (EM) simulation.

Optimal parameter set selection of algorithms

In optimization problems, the correct determination of the population size of the algorithm is one of the most important points. If this value is chosen smaller than it should be, the algorithm is most likely stuck at the local optimum. Likewise, when this value is selected large, it has handicaps such as waste of resources and high time, although the solution space is scanned very wide [46]. Although the main purpose of the study is to compare the objective functions, experiments were made with different population parameters in both algorithms in order to have the most optimal parameters of the algorithms used.

Performance comparisons were made for the cost₂ (17) function using OF₁₁–OF₁₂ (6–7), one of the objective function pairs determined by DEA and SPEA2. Figure 2 shows typical cost and Function evaluation number (FEN) variations with repetition of the best performance selected from 10 different studies for population = 30, 50, and 80. In addition, the cost and FEN variations in Fig. 2 are given in Table 2 as a numerical table. The FEN value in Fig. 2 gives information about the number of steps reached for the minimum and maximum column, and the total number of steps for the average column. The population value for DEA was chosen as 80 due to both its minimum cost and narrow maximum minimum range. Likewise, 50, which has the minimum and low average cost in SPEA2, was chosen as the optimum population value. These population values will be used in all subsequent parts of the study.

[See PDF for image]

Fig. 2

Typical cost₂ and FEN variations obtained with iteration the best performance selected from among 10 runs based on population parameter selection a DEA, b SPEA2

Table 2. Performance evaluations of algorithms by population parameter for results in Fig. 2 (maximum iteration = 30)

Population		DEA			SPEA2
		Minimum	Maximum	Mean	Minimum	Maximum	Mean
30	Cost2	0.1378	0.1827	0.1494	0.1273	0.1790	0.1466
30	FEN	300	60	930	630	60	930
50	Cost2	0.1313	0.1750	0.1393	0.1136	0.1638	0.1292
50	FEN	550	100	1550	950	100	1550
80	Cost2	0.1268	0.1560	0.1383	0.1260	0.1641	0.1350
80	FEN	1120	160	2480	2240	160	2480

Performances of objective functions

The objective function is the most important function that provides the best solution (maximum or minimum) consisting of decision variables and the parameters of these variables. For this reason, in this section, performance comparisons of 5 different types of objective functions adapted from mathematical models will be made. As we mentioned in the previous sections, the optimization problem may contain more than one objective function. For this purpose, DEA, which has a fixed-weighted objective function structure, and SPEA2, which has a variable-weighted objective function structure, were preferred separately for comparison.

Figure 3 shows typical cost₂ and FEN variations with repetition of the best performance selected from 10 different studies for the determined objective function pairs. In addition, the results corresponding to the cost₂ and FEN variations in Fig. 3 are given in Table 3 as a numerical table. In addition, the best results selected according to the cost₁ value are shown in the table without specifying the cost value. Because comparing the objective function pairs using the cost₁ value may be an erroneous approach. Of the results from the table and figure, OF₄₁–OF₄₂ (12–13) had the most unsuccessful results. Therefore, it is not included in Fig. 2. In addition, although the result with the highest S₁₁(dB) was obtained with OF₅₁–OF₅₂ (14–15), the most successful results in terms of both gain (dB) and S₁₁(dB) were found using OF₂₁–OF₂₂ (8–9) and OF₃₁–OF₃₂ (10–11).

Table 3. Performance evaluations of DEA and SPEA2 according to objective functions for optimization (maximum iterations = 30 and population (N) = 80–50)

Type (i = 1,2)	DEA (N = 80)					SPEA2 (N = 50)
	Cost1		Cost2			Cost1		Cost2
	Gain (dB)	S11 (dB)	Gain (dB)	S11 (dB)	Cost2	Gain (dB)	S11 (dB)	Gain (dB)	S11 (dB)	Cost2
OF1i	4.77	− 21.90	4.77	− 21.90	0.1276	4.70	− 29.35	4.70	− 29.35	0.1234
OF2i	5.65	− 13.55	5.65	− 13.55	0.1254	5.84	− 21.18	5.84	− 21.18	0.1092
OF3i	5.70	− 11.03	5.52	− 11.86	0.1327	5.33	− 10.84	4.63	− 36.71	0.1215
OF4i	3.45	− 0.01	4.06	− 14.95	0.1565	4.91	− 10.23	4.61	− 15.32	0.1412
OF5i	1.47	− 40.01	4.65	− 23.08	0.1292	2.73	− 46.49	4.73	− 29.02	0.1230

[See PDF for image]

Fig. 3

Typical cost₂ and FEN variations obtained with iteration the best performance selected from among 10 runs based on objective function selection a DEA, b SPEA2

Performance changes according to weight coefficient

Since both objective functions are tried to converge to zero on the basis of the working principles of the algorithms used, the weight coefficients within the objective function pairs are of great importance. Here, among the measurement functions chosen as decision variables, wc₁ determines the weight of the 30-degree directivity gain, and wc₂ determines the weight of S₁₁. In the parts of the study done so far, wc₁ = wc₂ = 0.5 was taken. In this section, we will see the results that the algorithms will find against different weight coefficients. Table 4 shows the measurement function results corresponding to the results with the lowest value according to cost₁ and cost₂ obtained with DEA. It is seen that the weight coefficients in OF₂₁–OF₂₂ (8–9) and OF₅₁–OF₅₂ (14–15) affect the results of the algorithm as expected. However, since OF₃₁–OF₃₂ (10–11) is an exponential model, it is seen that this situation is not reflected in the results too much. In Table 5, the measurement function results corresponding to the results with the lowest value according to cost₁ and cost₂ of the results found with SPEA2 are given numerically. Here, contrary to the situation in DEA, it is seen that the weight coefficients in all results affect the results of the algorithm as expected.

Table 4. Performance evaluations of DEA according to o the weight coefficients for optimization (maximum iterations = 30 and population (N) = 80) (i = 1, 2)

wc1	wc2	Cost1						Cost2
		OF2i		OF3i		OF5i		OF2i		OF3i		OF5i
		Gain (dB)	S11 (dB)	Gain (dB)	S11 (dB)	Gain (dB)	S11 (dB)	Gain (dB)	S11 (dB)	Gain (dB)	S11 (dB)	Gain (dB)	S11 (dB)
0.3	0.7	4.64	− 25.26	5.67	− 10.81	1.50	− 48.46	4.64	− 25.26	4.53	− 27.61	4.44	− 26.31
0.5	0.5	5.65	− 13.55	5.70	− 11.03	1.47	− 40.01	5.65	− 13.55	5.52	− 11.86	4.65	− 23.08
0.7	0.3	5.70	− 10.21	6.85	− 12.98	2.82	− 43.43	5.64	− 10.92	6.85	− 12.98	6.78	− 12.51

Table 5. Performance evaluations of SPEA2 according to o the weight coefficients for optimization (maximum iterations = 30 and population (N) = 50) (i = 1,2)

wc1	wc2	Cost1						Cost2
		OF2i		OF3i		OF5i		OF2i		OF3i		OF5i
		Gain (dB)	S11 (dB)	Gain (dB)	S11 (dB)	Gain (dB)	S11 (dB)	Gain (dB)	S11 (dB)	Gain (dB)	S11 (dB)	Gain (dB)	S1111 (dB)
0.3	0.7	4.75	− 24.26	5.69	− 11.23	3.38	− 51.84	4.66	− 26.25	4.65	− 47.87	4.28	− 42.62
0.5	0.5	5.84	− 21.18	5.33	− 10.84	2.73	− 46.49	5.84	− 21.18	4.63	− 36.71	4.73	− 29.02
0.7	0.3	5.71	− 11.02	5.21	− 10.58	4.05	− 52.56	5.71	− 11.02	5.18	− 11.91	5.63	− 11.68

Directivity and S₁₁ for antenna

Until this part of the study, we have completed all the steps that can be done to find the optimum results. Now, we will obtain the measurement functions selected as the decision variables of some of the results found by using the 3D EM simulation tool. Figures 4A and 5 show typical magnitude-frequency variations and directivity for DEA and some of the results from Table 4 in the previous section, respectively. In addition, the co-polarization and cross-polarization models are shown in Fig. 6. The effect of weight coefficients on the results can be seen more clearly in the graphs. Typical magnitude-frequency variations for a few selected among SPEA2 and the results from Table 5 in the previous section are shown in Fig. 4B and directivity is shown in Fig. 7. Additionally, Co-Polarization and Cross-polarization patterns are shown in Fig. 8.

[See PDF for image]

Fig. 4

S₁₁ of the antenna for different weight coefficient values selected from the results in Tables 4 and 5a DEA, b SPEA2

[See PDF for image]

Fig. 5

Directivity of antenna for results in Fig. 4A awc_1–2 = 0.3–0.7, bwc_1–2 = 0.5–0.5, cwc_1–2 = 0.7–0.3

[See PDF for image]

Fig. 6

Pattern of antenna for results in Fig. 4A a Co-Polarization, b Cross-Polarization, in E-plane (wc_1–2 = 0.7–0.3)

[See PDF for image]

Fig. 7

Directivity of antenna for results in Fig. 4B awc_1–2 = 0.3–0.7, bwc_1–2 = 0.5–0.5, cwc_1–2 = 0.7–0.3

[See PDF for image]

Fig. 8

Pattern of antenna for results in Fig. 4B a Co-Polarization, b Cross-Polarization, in E-plane (wc_1–2 = 0.5–0.5)

Discussion and future works

Good performance of an optimization depends on a properly selected objective function pair as well as balanced parameter settings. In many studies, the importance of the objective function pair is overlooked. In most studies, only one pair of objective functions is chosen, and this selection is often nonlinear [47]. However, the biggest source for optimization problems is mathematical methods. Many objective functions can be derived based on mathematical methods. In this study, five different nonlinear objective function pairs called polynomial, power, exponential, gaussian and fourier models were determined. The determined objective functions were tested on different algorithms and their performances were compared. These original models form the basis of the study. In addition to all these, modifications in the antenna model may lead to more effective solutions in order to obtain more effective results. Of course, determining effective objective functions is still an active area of research. In future studies, different models in objective function pairs can be cross-matched and tested. In addition, different models can be derived from mathematical methods.

Conclusions

In this study, objective function pairs, which are the most important actors in the success of multi-objective algorithms, were adapted in 5 different types (polynomial, power, exponential, gaussian and fourier) by using mathematical models and applied separately in DEA, which has a fixed-weighted objective function structure, and SPEA2, which has a variable-weighted objective function structure. In the study, quadrafilar helical antenna that can be used in the 3.5 GHz band was preferred as a model. Here, the 30-degree directivity gain and S₁₁ are selected from the measurement functions determined as the decision variable. When the study was considered multidimensional, the most unsuccessful results were found with OF₄₁–OF₄₂ (12–13) adapted from the gaussian model, and the successful results with OF₂₁–OF₂₂ (8–9), OF₃₁–OF₃₂ (10–11) and OF₅₁–OF₅₂ (14–15) adapted from the power, exponential and fourier models, respectively. It was seen that the most successful results were obtained with OF₂₁–OF₂₂ (8–9) adapted from the power model. It has been shown that the weight coefficients within the objective functions successfully fulfill their task. In addition, if we compare the algorithms with each other, it is seen that the cost values obtained from the SPEA2 algorithm are lower. In addition, some results obtained from the study were obtained by using the geometric design parameters of the selected antenna using 3D EM simulation tool, and the accuracy of the measurement functions selected as decision variables were confirmed. In the light of all this information, objective functions adapted from mathematical models are effects functions for creating objective functions in other design optimization problems to be made in the future. This proposed adaptation technique, which is also a novelty of the study, can be quickly adapted to any optimization problem.

Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

1. Gass, SI. Making decisions with precision; 2000; Business Week:

2. Murata, T., & Ishibuchi, H. (1995). MOGA: multiobjective genetic algorithms. In 2nd IEEE International Conference on Evolutionary Computation, (pp. 289–294)

3. Ishibuchi, H; Murata, T. A multi-objective genetic local search algorithm and its application to flowshop scheduling. IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews); 1998; 28, 3 pp. 392-403. [DOI: https://dx.doi.org/10.1109/5326.704576]

4. Fığlalı, A. (2008). Optimizasyona Giriş. Optimizasyon Seminerleri Dizisi, 2–27.

5. Morsy, M. M., & Harackiewicz, F. (2014). On optimization of the ground conductor of helical antennas. In IEEE Antennas and Propagation Society International Symposium (APSURSI), (pp. 45–46)

6. Kumar, BP; Kumar, C; Kumar, VS; Srinivasan, VV. Optimal radiation pattern of the element for a spherical phased array with hemispherical scan capability. IEEE Antennas and Wireless Propagation Letters; 2017; 16, pp. 2780-2782. [DOI: https://dx.doi.org/10.1109/LAWP.2017.2745541]

7. Ericsson, A; Sjöberg, D. A resonant circular polarization selective structure of closely spaced wire helices. Radio Science; 2015; 50, 8 pp. 804-812. [DOI: https://dx.doi.org/10.1002/2014RS005641]

8. Zhou, D., Gao, S., Abd-Alhameed, R. A., Zhang, C., Alkhambashi, M. S., & Xu, J. D. (2012). Design and optimisation of compact hybrid quadrifilar helical-spiral antenna in GPS applications using genetic algorithm. In 6th European Conference on Antennas and Propagation (EUCAP), (pp. 1–4)

9. Javan, H., & Estrella, A. (2009). Helix parameters optimization for Wi-Max applications. In 7th International Conference on Information, Communications and Signal Processing (ICICS)

10. Moghaddam, ES. Design of a printed quadrifilar-helical antenna on a dielectric cylinder by means of a genetic algorithm [antenna applications corner]. IEEE Antennas and Propagation Magazine; 2011; 53, 4 pp. 262-268. [DOI: https://dx.doi.org/10.1109/MAP.2011.6097348]

11. Özkan, A; Işık, Ş; Günkaya, Z; Özkan, K; Banar, M. Comparison of different modeling methods for prediction of palladium adsorption onto waste orange peel. Mühendislik Bilimleri ve Tasarım Dergisi; 2021; 9, 3 pp. 758-767. [DOI: https://dx.doi.org/10.21923/jesd.708225]

12. Zhang, J; Yang, J; Aydin, ME; Wu, JY. Mathematical modelling and comparisons of four heuristic optimization algorithms for WCDMA radio network planning. International Conference on Transparent Optical Networks; 2006; 3, pp. 253-257.

13. Nakamura, M., & Sakakibara, K. (2019). Formal verification and mathematical optimization for autonomous vehicle group controllers. In ACM/IEEE 22nd International Conference on Model Driven Engineering Languages and Systems Companion (MODELS-C), (pp. 732–733)

14. Ramli, M., Sundari, T., & Halfiani, V. (2018). Mathematical optimization model of parking capacity for parking area in triangular shape. In International Conference on Electrical Engineering and Informatics (ICELTICs), (pp. 164–167)

15. Lemeshko, O., Lebedenko, T., Mersni, A., & Hailan, A. M. (2019). Mathematical optimization model of congestion management, resource allocation and congestion avoidance on network routers. In International Conference on Information and Telecommunication Technologies and Radio Electronics (UkrMiCo).

16. Balanis, CA. Antenna theory: Analysis and design; 1997; 2 John Wiley and Sons, Inc.:

17. Kraus, JD. Antennas; 1988; 2 McGraw Hill, Inc.:

18. Srivastava, A. (2021). Design and implementation of quadrifilar helical antenna for 60 GHz band application. In 5th International Conference on Trends in Electronics and Informatics (ICOEI), (pp. 601–605)

19. Srivastava, A., Tiwari, R., CHT, K. T. Y., & Venkateswaran, K. (2020). Quadrifilar helix antenna for 28 GHz band application. In Second International Conference on Inventive Research in Computing Applications (ICIRCA), (pp. 858–861)

20. Shcherbyna, O., Tomai, O., & Kozhokhina, O. (2018). Quadrifilar helical antennas with different types of supply lines. In 2018 Advances in Wireless and Optical Communications (RTUWO), (pp. 167–170)

21. Yan, Y-D., Jiao, Y-C., Tian, R., & Zhang, W. (2018). A dual-band circularly-polarized Quadrifilar helix antenna. In 11th UK-Europe-China Workshop on Millimeter Waves and Terahertz Technologies (UCMMT)

22. Liu, X., Yao, S., Russo, N., & Georgakopoulos, S. (2018). Tri-band reconfigurable origami helical array. In IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (pp. 1231–1232)

23. Alsawaha, HW; Safaai-Jazi, A. Ultra-wideband hemispherical helical antennas. IEEE Transactions on Antennas and Propagation; 2010; 58, 10 pp. 3175-3181. [DOI: https://dx.doi.org/10.1109/TAP.2010.2055806]

24. Alsawaha, HW; Safaai-Jazi, A. A new technique for bandwidth improvement and size reduction of helical antennas. Microwave and Optical Technology Letters; 2011; 53, 12 pp. 2990-2994. [DOI: https://dx.doi.org/10.1002/mop.26427]

25. Barts, R. M., & Stutzman, W.L. (1997). A reduced size helical antenna. In IEEE Antennas and Propagation Society International Symposium, (pp. 1588–1591)

26. Safaai-Jazi, A; Cardoso, JC. Radiation characteristics of a spherical helical antenna. IEE Proc Microwave Antennas Propagation; 1996; 143, 1 pp. 7-12. [DOI: https://dx.doi.org/10.1049/ip-map:19960037]

27. Weeratumanoon, E; Safaai-Jazi, A. Truncated spherical helical antennas. IEEE Electronic Letters; 2000; 36, 7 pp. 607-609. [DOI: https://dx.doi.org/10.1049/el:20000449]

28. Tice, TE; Kraus, JD. The influence of conductor size on the properties of helical beam antennas. Proceedings of the IRE; 1949; 27, 1296.

29. King, H; Wong, J. Characteristics of 1 to 8 wavelength uniform helical antennas. Antennas and Propagation, IEEE Transactions; 1980; 28, 2 pp. 291-296. [DOI: https://dx.doi.org/10.1109/TAP.1980.1142322]

30. Wong, J; King, H. Empirical helix antenna design. Antennas and Propagation Society International Symposium; 1982; 20, 1 pp. 366-369.

31. Djordjevic, AR; Zajic, AG; Ilic, MM; Stuber, GL. Optimization of Helical antennas [Antenna Designer's Notebook]. IEEE Antennas and Propagation Magazine; 2006; 48, 6 pp. 107-115. [DOI: https://dx.doi.org/10.1109/MAP.2006.323359]

32. Qiu, X; Xu, J; Tan, KC; Abbass, HA. Adaptive cross- generation differential evolution operators for multiobjective optimization. IEEE Transactions on Evolutionary Computation; 2016; 20, 2 pp. 232-244. [DOI: https://dx.doi.org/10.1109/TEVC.2015.2433672]

33. Arindam, D; Jibendu, SR; Bhaskar, G. Performance comparison of differential evolution, particle swarm optimization and genetic algorithm in the design of circularly polarized microstrip antennas. IEEE Transactions on Antennas and Propagation; 2014; 62, 8 pp. 3920-3928. [DOI: https://dx.doi.org/10.1109/TAP.2014.2322880] 1371.78346

34. Liouane, H., Chiha, I., Douik, A., & Messaoud, H., (2012). Probabilistic differential evolution for optimal design of LQR weighting matrices. In 2012 Computational Intelligence for Measurement Systems and Applications (CIMSA)

35. Rocca, P; Oliveri, G; Massa, A. Differential evolution as applied to electromagnetics. IEEE Antennas and Propagation Magazine; 2011; 53, 4 pp. 169-171. [DOI: https://dx.doi.org/10.1109/MAP.2011.6097316]

36. Hongwen, Y; Xinran, L. A novel attribute reduction algorithm based improved differential evolution. Intelligent Systems (GCIS) Second WRI Global Congress on; 2010; 3, pp. 87-90.

37. Das, S; Abraham, A; Chakraborty, UK; Konar, A. Differential evolution with neighborhood based mutation operator. IEEE Transactions on Evolutionary Computation; 2009; 13, 3 pp. 526-553. [DOI: https://dx.doi.org/10.1109/TEVC.2008.2009457]

38. Ney, R. C., Canha, L. N., Adeyanju, O. M., & Arend, G. (2019). Multi-objective optimal planning of distributed energy resources using SPEA2 algorithms considering multi-agent participation. In 54th International Universities Power Engineering Conference (UPEC)

39. Huseyinov, I., & Bayrakdar, A. (2019). Performance evaluation of NSGA-III and SPEA2 in solving a multi-objective single-period multi-item inventory problem. In 4th International Conference on Computer Science and Engineering (UBMK), (pp. 531–535)

40. Zitzler, E., Laumanns, M., & Thiele, L. (2001). SPEA2: Improving the performance of the strength pareto evolutionary algorithm. In Technical Report 103, Computer Engineering and Communication Networks Lab (TLK), Swiss Federal Institute of Technology (ETH) Zurich.

41. Kim, M., Hiroyasu, T., & Miki, M. (2004). SPEA2+: Improving the performance of the strength Pareto evolutionary algorithm 2. In Parallel Problem Solving from Nature - PPSN VIII, (pp. 742–751).

42. Qin, Y; Ji, J; Liu, C. An entropy-based multiobjective evolutionary algorithm with an enhanced elite mechanism. Applied Computational Intelligence and Soft Computing; 2012; 2012, 2 17.

43. Ergül, E. U. (2011). An active elitism mechanism for multiobjective evolutionary algorithms. International Journal of Artificial Intelligence and Expert Systems, 2(4).

44. Michalewicz, Z. Genetic algorithms + data structures = evolution programs; 1994; Springer-Verlag: [DOI: https://dx.doi.org/10.1007/978-3-662-07418-3] 0818.68017

45. Chapra, SC; Canale, RP. Numerical methods for engineers; 2010; McGraw-Hill Higher Education:

46. Reeves, CR; Rowe, JE. Genetic algorithms: principles and perpectives; 2003; Kluwer Academic Publishers: 1029.90081

47. Belen, A; Tari, O; Mahouti, P; Belen, MA; Çalışkan, A. Surrogate based design optimization of multi-band antenna. Applied Computational Electromagnetics Society Journal.; 2022; 37, 1 pp. 33-40.