Joint user grouping and power control using whale optimization algorithm for NOMA uplink systems

Encircling prey

In this approach, the humpback-whales can locate the prey-location of the prey and en-circle that region. Considering that, the location of the optimal design in the search region is not known in the beginning. Hence, the algorithm WOA provides the best solution that is nearer to the optimal value. First determine the best solution regarding location and then change the position according to the current condition of the other search agents concerning to determine the best solution. Such an approach is described mathematically and can be expressed as: (16)${\displaystyle \vec{E}=|A.\overrightarrow{X^{\ast }}\left(t\right)-X\left(t\right)|}$ (17)${\displaystyle \vec{Y}\left(t\right)=\vec{X}\left(t+1\right)}$ (18)${\displaystyle \vec{Y}\left(t\right)=\overrightarrow{X^{\ast }}\left(t\right)-\vec{B}.\vec{E}}$

where, $\vec{B}$ and $\vec{A}$ represents the coefficients-vectors, t defines the initial iteration and X^∗ and $\vec{X}$ both describes the position- vector where X^∗ includes the best solution so far acquired. || and . defines the absolute and multiplication. Noted that the position vector X^∗ is updated for each iteration until to find the best solution. The coefficients vector vectors $\vec{B}$ and $\vec{A}$ can be determined as: (19)${\displaystyle \vec{B}=2\vec{b}.\vec{r}-\vec{b}}$ (20)${\displaystyle \vec{A}=2.\vec{r}}$

where $\vec{r}$ indicates random vector 0 ≤ r ≤ 1 and $\vec{b}$ represent a vector with a value between 2 and 0, which is decreasing linearly during the iteration.

Spiral bubble-net feeding maneuver

Two techniques are proposed to predict accurately the bubble-net activity of humpback-whales.

1. Shrinking en-circling

This type of techniques is achieved by decreasing the value of $\vec{b}$ using Eq. (19). It is to be noted that the variation range of $\vec{B}$ is also reduced by the value $\vec{b}$. Therefore, $\vec{B}$ is a random value from [ − b, b], where the value of b is decreasing from 2 to 0 during the iterations.

2. Spiral updating position

In spiral method, a relationship between the location of prey and whale to impersonate the helix-shaped operations is represented in the form of mathematical equation of humpback-whales in the following manner: (21)${\displaystyle \vec{Y}\left(t\right)={\vec{E}}^{{}^{\prime }}.e^{pq}.cos\left(2\pi q\right)+{\vec{X}}^{\ast }}$

where ${\vec{E}}^{{}^{\prime }}=|{\vec{X}}^{\ast }\left(t\right)-\vec{X}\left(t\right)|$, which represents the distance between prey and the i − th whale. l denotes the random number (−1 ≤ l ≤ 1).b represents logarithmic spiral, which is a constant number and . indicates the multiplication operation. It’s worth noting that humpback-whales swim in a shrinking-circle around their prey while still following a spiral-shaped direction. To predict this concurrent action, an equation is derived to represent the model can be expressed as: (22)${\displaystyle \vec{Y}\left(t\right)=\left\{\begin{array}{cc}{\vec{X}}^{\ast }\left(t\right)-\vec{E}.\vec{B},\hfill & \text{if}d<0.5\hfill \\ {\vec{E}}^{{}^{\prime }}.e^{pq}.cos\left(2\pi q\right)+{\vec{X}}^{\ast },\hfill & \text{if}d\ge 0.5\hfill \end{array}\right.}$

where d represents a random number (0 ≤ d ≤ 1). Further, the searching behaviour of humpback-whales for prey is randomly in the bubble-net approach. The following is the representation of mathematical model for bubble net approach.

Prey searching technique

To locate prey, same strategy based on the modification of the $\vec{B}$ vector can be utilized (exploration). In reality, humpback-whales search at random based on their location. As a result, we select $\vec{B}$ randomly with values $\vec{B}>1$ or $\vec{B}<-1$ to compel the search-agent to step away from a target value. Comparison with exploitation, modify the location of every search- agent in the sample space, based on randomly selected process until to obtained a better solution. This operation and $\left|\vec{B}\right|>1$ place an emphasis on the exploration phase and enable WOA to perform global-searching. This can be represented below: (23)${\displaystyle \vec{E}=|\vec{A}.\overrightarrow{X_{rand}}-\vec{X}|}$ (24)${\displaystyle \vec{Y}\left(t\right)=\overrightarrow{X_{rand}}-\vec{B}.\vec{E}}$

where $\overrightarrow{X_{rand}}$ indicates a position-vector that is randomly selected from the existing space.

The algorithm WOA comprised of a selection of random samples. For every iteration, the search agents change their locations in relation to either a randomly selected search agent or the best solution acquired so far in this. For both cases exploitation phase and exploration the value of b is decreasing in the range from 2 to 0 accordingly. As the value of $\left|\vec{B}\right|>1$, a randomly searching solution is selected, while the optimal solution is obtained when $\left|\vec{B}\right|<1$ for updating the search- location of the agents. Based on the parameter d, the WOA is used as a circular or spiral behaviour. Ultimately, the WOA is ended by the successful termination condition is met. Theoretically, it provides exploration and exploitation capability. Therefore, WOA can still be considered as a successful global optimizer. The WOA is described in Algorithm 1.

 
 Data: Set the input control variables 
    M,N,γm,{gm},{sn} 
  Population initialization X1,X2,.........Xn 
     Result: X∗ (Best search agent for user- pairing/grouping). 
    List all the users with decreasing order of Jm. 
    while  t < (total iterations) 
        for  every search user 
       Initialize b,B,A,q and d 
          if1(d < 0.5) 
            if2(|B| < 1) 
            Existing search user position is updated using equation (16) 
            else if2(|B|≥ 1) 
            Randomly selected a search user (Xrand) 
            Existing position of search user is updated using equation (23) 
            end if2 
         else if1(d ≥ 0.5) 
          Exiting position of search user is updated using equation (21) 
          end if1 
       end for 
       Examine the position of every search user in the search region if above the search 
    region then modify it. 
        Determine the position of every search user 
        If a best solution becomes available, update X* 
        t=t+1 
    end while 
    return X∗ 
                        Algorithm 1: WOA    

Social hierarchy

For mathematical representation of GWO, we assume that α is fittest alternative solution used to mimic the social hierarchy. As a result, the second and third best solutions are designated as β and δ, respectively. The remaining member approaches are now considered to be ω. For algorithm, the hunting (optimization) is led by α, β and δ. These three wolves are accompanied by the ω.

Encircling prey

Grey wolves encircle prey during the hunting. The following equations are presented to mathematical model the encircling actions. (25)${\displaystyle \vec{A}=|\vec{D}.\overrightarrow{X_s}\left(t\right)-\overrightarrow{X\left(t\right)}|}$ (26)${\displaystyle \vec{Y}\left(t\right)=\vec{X}\left(t+1\right)}$ (27)${\displaystyle \vec{Y}\left(t\right)=\overrightarrow{X_s}\left(t\right)-\vec{B}.\vec{A}}$

where $\vec{B}$ and $\vec{D}$ represents coefficient vectors, t is the exiting iteration, $\vec{X}$ and $\overrightarrow{X_s}$ defines the position vector of a grey wolf and prey.

The vectors and are computed in the following manner: (28)${\displaystyle \vec{B}=2.\vec{b}.\overrightarrow{d_1}-\vec{b}}$ (29)${\displaystyle \vec{D}=2.\overrightarrow{d_2}}$

where 0 ≤ d₁ ≤ 1 and 0 ≤ d₂ ≤ 1 indicates random vector and $\vec{b}$ component decreasing linearly over the entire iteration from 2 to 0.

Hunting

Grey wolves do have capability to detect and encircle prey. The α normally leads the chase. The β and δ can also engage in hunting. However, in an arbitrary search space, we have no idea that where is the optimal (prey) location. Hence, first acquired the three best solutions so far and then search other agents (containing ω’s) in accordance with the best search agent’s location. In this respect, the following equations are provided. (30)${\displaystyle \overrightarrow{A_{\alpha }}=|\overrightarrow{D_1}.\overrightarrow{X_{\alpha }}-\vec{X}|}$ (31)${\displaystyle \overrightarrow{A_{\beta }}=|\overrightarrow{D_2}.\overrightarrow{X_{\beta }}-\vec{X}|}$ (32)${\displaystyle \overrightarrow{A_{\delta }}=|\overrightarrow{D_3}.\overrightarrow{X_{\delta }}-\vec{X}|}$ (33)${\displaystyle \overrightarrow{X_a}=\overrightarrow{X_{\alpha }}-\overrightarrow{{\beta }_1}.\left(\overrightarrow{A_{\alpha }}\right)}$ (34)${\displaystyle \overrightarrow{X_b}=\overrightarrow{X_{\beta }}-\overrightarrow{{\beta }_2}.\left(\overrightarrow{A_{\beta }}\right)}$ (35)${\displaystyle \overrightarrow{X_c}=\overrightarrow{X_{\delta }}-\overrightarrow{{\beta }_3}.\left(\overrightarrow{A_{\delta }}\right)}$ (36)${\displaystyle \vec{Y}\left(t\right)=\frac{\overrightarrow{X_a}+\overrightarrow{X_b}+\overrightarrow{X_c}}{3}}$

Attacking prey

The wolves complete the chase by hitting the target once it cease running. The mathematical model of attacking prey approach can be achieve by decreasing the value of $\vec{b}$. It’s worth mentioning that the variance range of $\vec{B}$ is also limited by $\vec{b}$.

That is $\vec{B}$ in the range of [ − b, b], which is a random value and decreasing over the entire iteration from 2 to 0. If any random values ranges between [ − 1, 1], then new location of a search-agent lies between exiting and prey location.

Search for prey

Grey wolves primarily hunt depending on the locations of the α, β, and δ. At the starting they diverge from other wolves to hunt and then combine to hit prey. The mathematical model of divergence can be achieved by utilizing the value of $\vec{B}$. For divergence, random values of $\vec{B}<1$ or $\vec{B}>1$ is used by the search agent. This process enables the GWO algorithm to search globally. In nature, the D vector can even be assumed as the impact of barriers to pursuing prey. In general, natural barriers arise in wolves’ hunting paths and discourage them from approaching prey effectively and easily. This is precisely depend on the vector D. It will arbitrarily give the prey a weight to find it tougher and farther to catch for wolves, depending on location of the wolf, or likewise.

The suggested social hierarchy supports GWO algorithm in sustaining the best solutions achieved so far through iteration. By using hunting approach, it enables agents to search the likely location of prey. The GWO is described in Algorithm 2.

 
    Data: Set the input control variables 
    M,N,γm,{gm},{sn} 
  Population initialization X1,X2,.........Xn 
     Initialization of B,b and D 
    Result: Xα (Best search agent for user- pairing/grouping). 
    List all the users with decreasing order of Jm. 
    Determine fitness of every search agent 
    Xα,Xβ and Xδ. 
    while  t < (total iterations) 
        for  every search user 
            Update the existing location of search agent by using equation (36) 
              Update B, b and D 
            Determine fitness of search agent 
              UpdateXα, Xβand Xδ 
               t=t+1 
        end for 
    end while 
    return Xα 
                                             Algorithm 2: GWO