**CHAPTER 3 PROPOSED ALGORITHMS**

**3.1. PIGEON INSPIRED OPTIMIZATION**

Pigeons are parts of the society Columbiformes which covers doves and pigeons [121].

Pigeons were used for transmitting information by the Egyptians, including transpired during various fighting operations. Homing pigeons can get their places by using three homing mechanisms: magnetic field, sun, and landmarks [122].

**3.1.1 ** **OVERVIEW OF PIGEON INSPIRED OPTIMIZATION **

Extensive swarm intelligence analyses have shown how some animals, like mammals or fishes, communicate between them in the natural environments in swarm [123]. The range of these swarms in size start form small numbers living in natural places to organized colonies that held huge areas and contain millions of individuals. The skills of the group in swarms show a good rubustness and flexibility [124] like preparation of routing [125], constraction of nest [126], managing the task [89] and different additional complex behaviors combined in some swarms presented in [127, 128, 129].

The abilities may be very poor for individuals in the swarm, while behaviors of the group can appear in the complete swarm, like a flock of bird migration, exploring in Bee and ant colonies. To complete the task by individuals is hard, while it easily achieved by a swarm of animals. The researcher observed that the smart skills group are materialize by sets of individuals with normal skills through information transmission and swarm intelligence.

In general, swarm intelligence offers among models from some common responses of simple tools combining with themselves, including its circumstances’, that drives into every evolution from a logical operative global model[34] [130]. Information passing between representatives is obscure and little. They actualize the interaction between representatives within a dispersed way without a centralized restriction mechanism. In other terms, the whole swarm intelligence form is uncomplicated in real life [131].

The group colony-level performance of the swarm that originate from the communications becomes helpful during performing complicated purposes [132].

Through World Wars (First and Second), pigeons served essentially to the UK, American, German, French and Australian forces. The significant ability of homing by pigeons to utilize the mixing of the magnetic area, the sun, and find their route around in the landmarks. Pigeons reasonably use various navigational tools through various elements presented by Guilford argues [133]. The mathematical model developed by Guilford and his partners for predicting when pigeons order the change of the route

from one to another. If pigeons begin the journey, they rely on extra tools. If during the campaign, they should turn on using landmarks if it needs to reassess the way and perform improvements.

Investigations show that the ability of a pigeon can detect various magnetic area proves that the pigeons’ compelling experiences of homing based on little magnetic bits in its beak. The beak of pigeons has iron crystals; they provide a nose of the birds to the north. Investigations explain that this appears to produce a mode within flags of magnetite bits transmitted into the mind through particular trigeminal nerves [134].

Additionally, the pigeon navigation is based also to the sun either entirely or partly, the highest of the sun help the pigeon to recognize the current location and home base [135]. Modern investigations about the behaviour of the pigeon further confirm the ability of the pigeon to recognize any landmarks, principal ways, rivers, and routes to the target straight.

**3.1.2 ** **MATHEMATICAL MODEL OF PIO **

About pigeons homing, a pair of processes have proposed using several rules [132]:

A. The operator of the Compass and Map: the ability of pigeons to sense the range of earth by utilizing magneto response to configure a map in their minds. It considers elevation from the sun and a compass for adjusting the path. Since they fly to their target, this operation less depended on the magnetic bit and sun.

B. The operator of the landmark: if the pigeons fly nearest to the target, they should depend on the landmarks near them. While it's common among landmarks, they must fly into the target straightly. While it’s far in the goal also unfamiliar of the landmarks, they must keep track of the closer pigeons among those landmarks.

**3.1.2.1 MAP AND COMPASS OPERATOR **

Within the operator of the compass and map, the location Pi and the speed Vi of pigeon
i are set and updated in D-dimension of search range within every iteration. The
following formula can measure the new location Pi and speed Vi of pigeon i in the t^{th}
repetition:

𝑉𝑖(𝑡)= 𝑉𝑖(𝑡 − 1). 𝑒^{−𝑅𝑡 }+rand.(Pg-Pi(t-1)) Equation 3-1
𝑃𝑖(𝑡) = 𝑃𝑖(𝑡 − 1) + 𝑉𝑖(𝑡) Equation 3-2

where R is the factor of compass and map, default random number is a rand, in the current location Pg is the best global, and which can achieve by associating each location with every pigeon. Figure 3.1 shows the process of compass and map form of PIO [132].

As presented in Figure 3.2, the better locations of every pigeon produced by using a compass and map. Through analysing all the locations of pigeons, that means the best

Figure 3.1. Map and compass operator model of PIO [132]

Figure 3.2 Lamndmark operator model [132]

location of the pigeon that is the right-centred. Adjusting the fly direction for every pigeon is through following this particular pigeon’s direction, which is based on Equation (3-2).

**3.1.2.2 LANDMARK OPERATOR **

Within the operator of the landmark, the number of pigeons decreased by half through
Np within each iteration. Still, the target is not near the pigeons; also it's unknown
among this landmark. Let the middle at a t^{th} repetition of some pigeon is Pc(t), also
assume each pigeon can fly to the target straightly. The location can update the pigeon
i at a t^{th} iteration as: [132]

𝑁𝑝 = 𝑐𝑒𝑖𝑙^{𝑁𝑝(𝑡−1)}

2 Equation 3-3

𝑃𝑖(𝑡)= 𝑃𝑖(𝑡 − 1)+ 𝑟𝑎𝑛𝑑(𝑃𝑐(𝑡)− 𝑃𝑖(𝑡 − 1)) Equation 3-4* *
where Pc(t) denotes the center position at the t^{th} iteration, defined as

𝑃𝑐(𝑡)=^{∑}𝑃𝑖(𝑡).𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝑃𝑖(𝑡))

𝑁𝑝∑𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝑃𝑖(𝑡)) * * * * * *Equation 3-5* *
To decrease the optimization difficulties into minimum, we select the fitness Pi (t)) =
1/(fmin(Pi (t)) + ε). For increasing the optimization into maximum, we select (Pi (t)) =
fmax(Pi (t)). For any individual pigeon, the optimal location of the Nc^{th} repetition is
identified among Pp, and Pp=min (Pi1, Pi2, …, PiNc).

As presented in Figure 3.2, the midpoint from all pigeons (The pigeon in the center of the range) is the goal in every repetition. Half from whole those pigeons (those pigeons outside from the circle) that are far away of the target should keep track of the pigeons near to the goal. The pigeons near the target (the pigeons inside the circle) should be able to fly to the destination fast.