Ant lion optimization

Mirjalili explains the ALO algorithm in depth in [Reference Mirjalili43], so the detailed description is excluded here. The interactions of the antlions and ants, demonstrated in the algorithm include the following rules [Reference Mirjalili43]:

1. 1. Ants and antlions are the search agents, and the ants change their positions in random directions all over the search space.

2. 2. Traps by antlions have impacts on the movements of the ants.

3. 3. Traps are constructed in proportion to the fitness of antlions. Higher is the fitness, larger is the trap, and higher is the possibility of catching the ants.

4. 4. In each iteration, an antlion and the most fitting antlion (elite) can capture each ant.

5. 5. The random walk adaptively decreases, and the probability of achieving a solution increases by increasing the sliding of ants to the antlions.

6. 6. An ant is hunted by the antlion when the ant is fit. This indicates that an anticipated solution is found.

7. 7. Antlion relocates itself to the recent hunting position and constructs a conical hole to optimize the ability to catch another ant.

The stochastic movements of the ants, throughout the optimization process, are defined as:

(1)$$\eqalign{R_i & = [ 0, \;\;cums( {2\gamma ( {t_1} ) -1} ) , \;\;\ldots , \;\;cums( {2\gamma ( {t_n} ) -1} ) , \;\;\ldots , \cr & \;\;cums( {2\gamma ( {t_M} ) -1} ) , \;}$$

where cums computes the cumulative sum. Parameter t _n is the iteration number and M is the maximum number of iterations. γ(t) is a random function described as follows:

(2)$$\gamma ( t ) = \left\{{\matrix{ {1\;\;\;\;\;{\rm if}\;rand > 0.5\;} \cr {\!\!\!0\;\;\;\;{\rm \,if}\;rand \le 0.5} \cr } } \right., \;$$

where rand within the interval [0, 1] is a random number.

At every step, ants update their locations on the basis of equation (1). The random walks are modified based on normalization using the following equation:

(3)$$R_{i\_norm} = \left({\displaystyle{{R_i-a_i} \over {b_i-a_i}}} \right)\times ( {d_i^t -c_i^t } ) + c_i^t , \;$$

where a _i is the minimum value and b _i is the maximum value of R _i. $c_i^t$ is the minimum value and $d_i^t$ is the maximum value of R _i at the tth iteration.

The hunting ability of an antlion is modelled through a roulette wheel operator. The roulette wheel chooses an antlion and assumes that ants are captured only by the selected antlion. Movement of the ants is affected by the position of the antlions. This is described by the following equations:

(4)$$c_i = c^t + Al_j^t , \;$$

(5)$$d_i = d^t + Al_j^t , \;$$

where c _i, d _i are the minimum and maximum values among all random walks for the ith ant. c ^t, d ^t indicate the minimum and maximum values among all random walks at the tth iteration. $Al_j^t$ represents the position of the chosen jth antlion at the tth iteration.

When the ants slid into the trap, they try to escape from it. On the other hand, the sliding of ants towards the antlions is updated adaptively. Hence the values of c ^t and d ^t are updated using the following equations:

(6)$$c^t = \displaystyle{{c^t} \over {{10}^w \times ( {t/n} ) }}, \;$$

(7)$$d^t = \displaystyle{{d^t} \over {{10}^w \times ( {t/n} ) }}, \;$$

where w is a constant determined according to the present iteration. w = 2 at t > 0.1T, w = 3 at t > 0.5T, w = 4 at t > 0.75T, w = 5 at t > 0.9T, and w = 6 at t > 0.95T.

An antlion also updates the hunting position to increase its chance of capturing new ant. This phenomenon is demonstrated in the following equation:

(8)$$Al_j^t = A_i^t \;{\rm if}\;{\rm fitness}\;{\rm of}\;A_i^t > {\rm fitness}\;{\rm of}\;Al_j^t , \;$$

where $A_i^t$ = position of the ith ant at the tth iteration.

The best antlion in the optimization cycle is considered as the elite. During the optimization process, the elite affects the random walks of all ants. Hence the position of each ant is updated by the simultaneous effect of the elite and the roulette wheel. This is expressed in equation (9) as:

(9)$$A_i^t = \displaystyle{{X_{al}^t + X_e^t } \over 2}, \;$$

where $X_{al}^t$ and $X_e^t$ represent the random walk nearby the antlion selected through a roulette wheel and nearby the elite obtained within the optimization process at the tth iteration, respectively.

Modification of ALO

The advantage of ALO is that it avoids optimal local values. In the original ALO, CPSO is integrated to improve the optimization functionality to provide more accurate design parameters. The working procedure of CPSO is focused on the behavior of searching for foods by a group of birds or schooling of fish. The searching agents (birds/fishes) search for optimal solutions in adequate space and thus increase the probability of obtaining optimum values. Each agent moves randomly towards a global best position with its velocity and position vectors which are updated by equations (10) and (11) [Reference Shi and Eberhart45] as:

(10)$$v_i = w \times v_i + c_1 \times rand_1 \times ( {\,p_i-x_i} ) + c_2 \times rand_2 \times ( {p_g-x_i} ) , \;$$

(11)$$x_i = x_i + v_i.$$

The variables of equation (10) are well-known in the domain of optimization through PSO and are described in [Reference Shi and Eberhart45]. The traditional PSO algorithm sometimes trapped prematurely in local optimums, which is a shortcoming. As described in [Reference Alatas, Akin and Ozer46], chaotic mappings help it to escape from the local optima. Chaotic maps constructed by mutating its initial state are apparently random deterministic and reproducible sequences. Several chaotic mappings, such as a logistic, tent, sinusoidal iterator, Gauss, etc. are adopted to enhance the global convergence of PSO. Here the CPSO considers the chaotic mutation operation based on logistic mapping to obtain the best optimal values. Logistic mapping shows vital usefulness in the improvement of the searching process. Mapping is employed in each iteration, and the position can be further updated using the formula as [Reference Alatas, Akin and Ozer46]:

(12)$$x_i( d ) = 4 \times x_{i-1}( d ) \times ( {1-x_{i-1}( d ) } ) , \;$$

where i is the current iteration. d = 1, 2, …, dim and dim is the dimension of searching range. The x _i is converted to the real positions as:

(13)$$x_i( d ) = lb( d ) + x_i \times ( {ub( d ) -lb( d ) } ) , \;$$

where lb is the lower bound and ub is the upper bound of the particles.

The optimization process of MALO involves the searching and updating strategy of both ALO and CPSO. In each iteration, the ALO explores the searching space first, by its search agents, the ants and the antlions. The positions of antlions define the optimizing parameters. After the exploration of searching space using ALO, the positions of antlions are updated and optimized employing CPSO. The positions of best-fitted antlions (elites) in each iteration are kept in memory. After the final iteration, the best elites are selected and the corresponding positions yield the global optimum parameters. The operational flow diagram of the MALO is shown in Fig. 1. The steps involved in the optimization process are as follows:

• Step 1: The positions of antlions and ants are initialized randomly.

• Step 2: The fitnesses of the antlions are calculated and the elite is selected.

• Step 3: An antlion is selected utilizing the roulette wheel. The random walks nearby the elite and the selected antlion are calculated.

• Step 4: The positions of the ants are updated using equations (1)–(7) and (9).

• Step 5: The positions of the antlions are updated using equation (8).

• Step 6: The fitnesses of the updated antlions are calculated and compared with that of the elite. If the fitness of the updated antlion is better, then it becomes the elite.

• Step 7: Better antlions are searched employing CPSO using equations (10) and (11).

• Step 8: The elite is updated using equations (12) and (13).

• Step 9: Steps 3–8 are repeated until the final iteration is reached.

Fig. 1. Flowchart of the MALO.

Validation of MALO algorithm

The proposed algorithm is validated by verifying its performance on five benchmark functions (two unimodal (single-peak) functions and three multimodal (multi-peak) functions). Table 1 shows the benchmark functions taken from [Reference Mirjalili43]. MALO is applied to optimize the above five functions by executing each for 30 runs with 30 dimensions and 1000 iterations per each term. Performance indices (mean and standard deviations) of the obtained minimums are computed for each function. The comparisons of the performance indices obtained using MALO and other algorithms, such as ALO, PSO, FPA, CS, FA, BA, and GA in [Reference Mirjalili43] are shown in Table 2. From the results shown in Table 2, it is worth noting that the performance of MALO is superior than other algorithms.

Table 1. Benchmark functions [Reference Mirjalili43].

Table 2. Mean (μ) and standard deviation (σ) of the benchmark functions and their comparison with other algorithms in [Reference Mirjalili43].

The values in bold indicate that they belong to our method.

The characteristics of the benchmark functions and their resultant convergence curves are plotted in Figs 2–6. In [Reference Mirjalili43], it is observed that the performance of ALO is better than various other optimization algorithms. The performance is evaluated by comparing the convergence characteristics of ALO with that of the other algorithms. As MALO is the modification of ALO, the convergence characteristics of this are compared with that of ALO and PSO instead of all other algorithms. The comparison with other algorithms is deliberately excluded, as it is already available in [Reference Mirjalili43]. The characteristics of the two unimodal test functions F₁ and F₂ are shown in Figs 2 and 3, respectively. These figures depict the benefits of high exploitation behavior of MALO. High exploitation enables the MALO algorithm to converge rapidly towards the optimum. Figures 4–6 show the characteristics of the multimodal functions F₃, F₄, and F₅, respectively. From these characteristics, it is clearly observed that the exploration ability of MALO is also high. This high level of exploring ability enables MALO to explore the desirable search domain. Due to the high level of exploration and exploitation ability of MALO the local optima or premature convergence are avoided and the global optima are attempted. Attempting the global optima by MALO can be clearly observed from the convergence curve of each benchmark function. The curve illustrates that the MALO exhibits superior convergence functionality, though the simulation time is a little longer than that of PSO. The simulation time for ALO is also longer than PSO and is closer to MALO. The time in case of MALO is longer, because of the hybridization of two algorithms. Embedding one algorithm (CPSO) to another (ALO) makes the hybrid algorithm (MALO) to some extent complex as compared to the individual one. That complexity in the new modified algorithm makes the simulation time longer. The simulation time of the algorithms PSO, ALO, and MALO for 1000 iterations is presented in Table 3. In spite of the long run time, the higher convergence functionality and ability to avoid local optima and attempting global optima makes use of the proposed MALO algorithm for antenna array synthesis and beamforming in smart antenna technology.

Fig. 2. (a) Function F₁. (b) Convergence curve of function F₁.

Fig. 3. (a) Function F₂. (b) Convergence curve of function F₂.

Fig. 4. (a) Function F₃. (b) Convergence curve of function F₃.

Fig. 5. (a) Function F₄. (b) Convergence curve of function F_4.

Fig. 6. (a) Function F₅. (b) Convergence curve of function F₅.

Table 3. Simulation time (min) of the algorithms for 1000 iterations: PSO, ALO, and MALO.

The values in bold indicate that they belong to our method.

Example-I: amplitude optimization

The use of MALO for the synthesis of two linear arrays (i.e. Example-Ia: 2N = 10 elements array and Example-Ib: 2N = 16 elements array) is considered here. The objective of this example is to obtain a Chebyshev array pattern as well as the peak SLL suppression. Peak SLL in a particular region can be minimized by considering the objective function formulated as:

(16)$$FObj = {\rm min}( {{\rm max}( {20{\rm log}\vert {AF( \phi ) } \vert } ) } ) $$

where max(20log|AF(ϕ)|) provides the peak SLL. Considering equation (16) as the objective function for MALO, the current amplitude excitations of the antenna elements are optimized. The search region for the current amplitudes is spread from 0 to 1. Putting the optimal values found through MALO in equation (14), AF(ϕ) for the said linear arrays with suppressed peak SLL is obtained. In this example, the positions of each element and their phases are kept fixed (i.e. x _n = 0.5λ and φ_n = 0⁰). The SLLs are suppressed in the regions, ϕ = [0⁰, 76⁰] and ϕ = [104⁰, 180⁰].

Example-II: position optimization

In this example, MALO is implemented for the optimization of the positions of the elements from the origin, as shown in Fig. 7. A 2 N =10 elements non-uniformly spaced linear array is considered as Example-IIa. It is optimized to take the same objective function in equation (16). For a uniform array of 0.5λ spaced elements the total length is 4.5λ. Therefore, the last elements' positions are fixed at 2.25λ on each side of the y-axis. The minimum spacing between two consecutive elements is taken as 0.25λ. Similar to Example-I, peak SLL suppressed AF(ϕ) is obtained by putting the optimum position values found through MALO in equation (14). Here the amplitudes of the elements and their phases are kept constant (i.e. I _n = 1 and φ _n = 0⁰). The SLL regions for this example are defined by ϕ = [0⁰, 76⁰] and ϕ = [104⁰, 180⁰].

In a smart antenna-based communication system, there are many applications that need minimization of CISLL adjacent to the main lobe. This can also be achieved by the elements position optimization. For this, the objective function is taken as [Reference Khodier and Al-Aqeel13]:

(17)$$FObj = \min [ {\propto_1{\rm max}\{ {20{\rm log}\vert {AF( {\phi_{AS}} ) } \vert } \} + \propto_2{\rm max}\{ {20{\rm log}\vert {AF( {\phi_{NS}} ) } \vert } \} } ] $$

where ϕ _AS are the SLL regions the same as Example-IIa defined by $\{ {[ {0{\rm^\circ }, \;{\rm \;}76{\rm^\circ }} ] \;{\rm and}\;[ {104{\rm^\circ }, \;{\rm \;}180{\rm^\circ }} ] } \}$. ϕ _NS represents the CISLL region, $\{ {[ {69{\rm^\circ }, \;{\rm \;}76{\rm^\circ }} ] \;\;{\rm and}\;[ {104{\rm^\circ }, \;{\rm \;}111{\rm^\circ }} ] } \}$. ∝₁ and  ∝ ₂ are constants with values 1 and 2, respectively.

Keeping the same conventions, the 10-elements linear array is again optimized to take the objective function as in equation (17) to realize the CISLL suppressed AF(ϕ). This is referred to as Example-IIb.

Example-III: phase optimization for beam steering

Nowadays, wireless and mobile technologies are widely applied in various wireless monitoring applications. Some of the wireless monitoring applications are water level and its quality monitoring, health monitoring of bridge pillar, building wall structures [Reference Castorina, Donato, Morabito, Isernia and Sorbello47, Reference Mauro, Castorina, Morabito, Donato and Sorbello48]. These wireless monitoring applications include a set of omnidirectional and directional antenna systems. The omnidirectional antenna transmits information to the distant terminal units. The directional antenna system collects the data from the antennas embedded in different places on the concrete structures. The antenna system adopts the beam-steering process for the data collection from the remote embedded antennas. Beam steering is the procedure of guiding an antenna array's main beam towards a specific direction.

Beam-steering application by the implementation of MALO in the linear array is demonstrated in this example. The simplest method of achieving beam steering for a linear array is through the optimization of phases of the elements. The positions of the elements and their amplitudes are kept fixed (i.e. x _n = 0.5λ and I _n = 1). Keeping the phase of the first element fixed at 0^° as a reference, optimal phases for rest of the elements are found by MALO and considering the AF(ϕ) as [Reference Khodier and Al-Aqeel13]:

(18)$$AF( \phi ) = \mathop \sum \limits_{n = 2}^{2N} {\rm exp}( {\,j[ {n\pi \cos ( \phi ) + \varphi_n} ] } ) + 1.$$

The SLL regions are defined by $\phi = [ {0{\rm^\circ }, \;\;{( {\phi_s-( {\Delta \phi_s/2} ) } ) }^0} ]$ and $\phi = [ {{( {\phi_s + ( {\Delta \phi_s/2} ) } ) }^0, \;\;180{\rm^\circ }} ]$. ϕ _s represents the steering angle and it lies in the band Δϕ _s. In this example, phases of a 20-elements linear array are optimized for steering angle ϕ _s = 45^° and the band of steering angle Δϕ _s = 14^°. The AF(ϕ), as in equation (18) with reduced SLL is obtained by putting the optimum phase values found applying MALO.

Example-IV: beam steering and null positioning by simultaneous optimization of amplitude and phase

In this example, MALO is employed for simultaneous optimization of amplitude and phase for smart antenna technology. The smart antenna application involves steering the main beam along with null positioning in the specific directions. The steering angle is characterized as the direction of the SOI angle, and the angles of nulls are characterized as SNOI angles. The beamforming here is achieved by an adaptive process. Figure 8 shows the block diagram of an adaptive array. Elements of antenna array receive the combined signal of subscriber's signal at the SOI angle and the interfering signals at SNOI angles. The combined signal is then multiplied with a complex-valued weight having the elements amplitude weights and the phase weights. These amplitude and phase weights are optimized in order to obtain the optimum output signal. Considering the self-developed MATLAB code, the main beam of the radiation pattern is directed at any specific SOI angle along with the placement of one or more nulls at any SNOI angles. In this section, a 20-elements array is considered to steer the main beam at the SOI angle ϕ _s = 45^° and nulls at SNOI angles ϕ _nl1 = 25^° and ϕ _nl2 = 65^°. The AF(ϕ) with reduced SLL is obtained by the optimum amplitude and phase excitations values found applying MALO. AF(ϕ) for this case is formulated as [Reference Balanis1]:

(19)$$AF( \phi ) = \mathop \sum \limits_{n = 1}^N w_n{\rm exp}( {j[ {( {n-1} ) kx_n\sin ( \phi ) + \varphi_n} ] } ) , \;$$

where w _n is the complex weight of the nth element expressed as a _nexp(jb _n). The amplitude weight of the nth element is a _n, and b _n is its phase weight.

Fig. 8. Adaptive antenna array geometry.