Final report aaa 2

FINAL REPORT ANALYSIS OF ALGORITHM 2019
Analysis of Algorithm By Faheem Ahmed Page 1
Cuckoo search
Cuckoo Search is basically an optimization algorithm that was implemented by Xin-she Yang
and Suash Debin the year 2009. It is inspired from nature’s behaviourto solve certain
optimization problems based on emcee birds, which can grab immediate dispute with the
encroachingcuckoos. It was developed based on the inspiration gained by the feature called
obligate brood parasitism in which few species of cuckoo birds will deploy the eggs in the emcee
bird’s nest ( basically of other species). Considering, if an emcee bird identifies that the eggs are
not its own, then it can pitch those unfamiliar eggs treating them as unwanted or simply vacate
their own nest and build a fresh nest in another place. Few of the cuckoo bird species like the
brood parasitic taper have emitted in a style, in which the female cuckoo birds are very habitual
in mimicry gauge of the eggs and in colors of certain opted host species. Cuckoo search
glamorized such reproduction behavior, and hence can be appeal for variety of optimization
problems. Once the first cuckoo egg opens and chick hatches out, its initial inclining activity is
touts the emcee eggs by furiously forcing the eggs to get out of the nest. This action influences
more number of cuckoo chicks for sharing food furnished by its emcee bird. Moreover, several
studies have shown that a baby
Cuckoo can also simulate the way of calling of the emcee chicks to acquire more ingress for the
feedingOccasion.
Cuckoos are fascinating birds, not only because of the beautiful sounds they can make, but also
because of their aggressive reproduction strategy. Some species such as the Ani andGuira
cuckoos lay their eggs in communal nests, though they may remove others’ eggs to increase the
hatching probability of their own eggs. Quite a number of species engage the obligate brood
parasitism by laying their eggs in the nests of other host birds (often other species). Some host
birds can engage direct conflict with the intruding cuckoos. If a host bird discovers the eggs are
not their owns, they will either throw these alien eggs away or simply abandon its nest and build
a new nest elsewhere. Some cuckoo species have evolved in sucha way that female parasitic
cuckoos are often very specialized in the mimicry in colors andpattern of the eggs of a few
chosen host species. This reduces the probability of theireggs being abandoned and thus
increases their reproductively.

For simplicity to describe CS algorithm, the following three idealized rules are used:
(1)Each cuckoo lays one egg at a time, and dumps its egg in randomly chosen nest;
(2) The nextgenerations carry the best nests with high quality of eggs;
(3) The number of available hostnests is fixed, and the egg laid by a cuckoo is discovered by the
host bird with a probability (pa).
The probability lies in the range of [0, 1]. In this case, the host bird can either throwthe egg away
or abandon the nest, and build a completely new nest. For simplicity, this lastassumption can be
approximated by the fraction probability (pa) of the n nests are replacedby new nests (with new
random solutions). For a maximization problem, the quality or fitness of a solution can simply be
proportional to the value of the objective function [25].
Based on the above three rules, the basic steps of the CS can be summarized as the pseudo code
shown in Fig.1.
Figure 2 is the flowchart diagram, which shows the main steps of the CS algorithm. Here the
concept of fitness, Fi is used to guide the Lévy flights during the search for the optimum nest
(solutions) in the N-dimensional search space. On the other hand, various studies have shown
that flight behavior of many animals and insects has demonstrated the typical characteristics of
Lévy flights [27–30]. Lévy flight is defined as a random walk with the step-lengths based on
heavy-tailed probability distributions. Studies on human behavior such as the Ju/’hoansi hunter-
gatherer foraging patterns also show the typical feature of Lévy flights. Subsequently, such
behavior has been applied to optimization and optimal search with promising capability [31, 32].
When generating new solutions x (t+1), for, say, a cuckoo i, a Lévy flight is performed as
Shown in Eq. 1
Xt+1i= xti+ α ⊕ L ´evy (λ) (1)
Whereα >0 is the step size which should be related to the scales of the problem of interests.
In most cases, we can use α = 1. The above equation is essentially the stochastic equation

for random walk. In general, a random walk is a Markov chain whose next status/location only
depends on the current location (the first term in the above equation) and the transition
probability (the second term). The product means entry wise multiplications. This entry wise
product is similar to those used in PSO, but here the random walk via Lévy flight is more
efficient in exploring the search space as its step length is much longer in the long run [33].
The Lévy flight essentially provides a random walk while the random step length is drawn from
a Levy distribution according to Eq. (2)
L ´evy ∼ u = t−λ (1 < λ ≤ 3) (2)
Which has an infinite variance with an infinite mean. Here the steps essentially form a random
walk process with a power law step-length distribution with a heavy tail. Some of the new
solutions should be generated by Lévy walk around the best solution obtained so far, this will
speed up the local search. However, a substantial fraction of the new solutions should be
generated by far field randomization and whose locations should be far enough from the current
best solution, this will make sure the system will not be trapped in a local optimum.
Characteristicsofcuckoo search:
 Every egg within the nest constitutes absolution, and an egg of cuckoo species depicts
fresh solution.
 The point here is to enlist fresh and prospectively best results to replace well-
fornothingsolutions in the nests.
 Having one egg in each nest is the easiest appearance in this mode.
 The algorithm can also be prolonged for more composite problems by considering
multiple eggs in the nest constitute a set of solutions.

Three idealized rules of cuckoo search:
 At most, each cuckoo bird can lay only one egg at a time, and place its egg in an
arbitrarily selected nest.
 The excellent nests which have the eggs with high caliber will fetch to the further
generation.
 The number of obtainable emcee’s nests is finalized, and the egg depicted by a cuckoo is
spotted by the emcee bird with the probability (0, 1).
Advantages:
 Concerned with multiple benchmarkoptimization problems
 Aims to boost up convergence
 Simplicity
 It can still be combined with other swarmbased algorithms
Algorithm of cuckoo search

Flow chart of cuckoo search

VARIANTS OF
CUCKOO SEARCH

Adaptive Cuckoo SearchAlgorithm
Adaptive CS algorithm, which can be used to address the structural engineering problems. The
structural engineering problems can be stated as, safety and reliability, production cost, stability,
sustainability and also rigidity features of the building structures. The authors have tried to
address few of these issues by proposing adaptive cuckoo search algorithm, which is an extended
version of cuckoo search algorithm that utilizes a strategy of adaptive step size selection and
diversification process [2]. This approach simultaneously preserves the balance of intensification
and diversification performances of CSA. In this work, they have demonstrated the effectiveness
of ACSA, with three structural engineering problems, Gear train design, Pressure Vessel design
and Three-Bar Truss design. With this, they have proved that, the structural optimization
problems can be easily solved by adaptive cuckoo search algorithm and several benchmark
structural engineering problems are validated. Finally, the comparative study of ACSA with
several other approaches proved the better searching ability.
The step size 𝛼, which manages the local and global searching, is assigned as constant in the
standard CSA, where𝛼 = 1 is applied. In this present work, a new adaptive cuckoo search
algorithm (ACSA) is presented. Instead of using constant value, the step size 𝛼 is adjusted
adaptively in the proposed ACSA, based on the assumption that the cuckoos lay their eggs at the
area with a higher egg survival rate. In this regard, by adjusting the step size 𝛼 adaptively, the
cuckoos search around the current good solutions for laying eggs this region probably will
contain the optimal solutions, and, on the contrary, they explore more rigorously for a better
environment if the current habitat is not suitable for breeding. The step size 𝛼 is determined
adaptively as follows:
Where𝛼𝐿is the predefined minimum step size, 𝛼𝑈is the predefined maximum step size, 𝐹𝑗is the
fitness value of the𝑗the cuckoo egg, and 𝐹min and 𝐹avg denote the minimum and the average

fitness values of all host nests, respectively. The flow of the ACSA is given in Algorithm 2.The
step size 𝛼determines how far a new cuckoo egg is located from the current host nest. Specifying
the minimum and the maximum L´evy flight step size values properly is crucial such that the
search process in the ACSA is neither too aggressive nor too ineffective. The 𝛼𝐿and 𝛼𝑀are
chosen based on the domain of x𝑖.
To evaluate the feasibility of the proposed ACSA, the algorithm is applied to optimize the five
benchmark functions with known global optima, where two of which are uni modaland three of
which are multimodal. The optimization4 The Scientific WorldJournal performance is compared
with the standard CSA. For each test function, the initial populations of 20 host nests are
generated randomly. For both the CSA and ACSA, the Euclidean distance from the known
global minimum to the location of the best host nest with the lowest fitness value is evaluated in
each iteration. The optimization4 The Scientific World Journal performance is compared with
the standard CSA. For each test function, the initial populations of 20 host nests are generated
randomly. The simulations are performed for 30independent runs. The optimization process
stops if the best fitness value is less than a given tolerance 𝜉 ≤ 10−5. For both the CSA and
ACSA, the Euclidean distance from the known global minimum to the location of the best host
nest with the lowest fitness value is evaluated in each iteration. The optimization4 The Scientific
World Journal performance is compared with the standard CSA. For each test function, the initial
populations of 20 host nests are generated randomly. The simulations are performed for
30independent runs. The Euclidean distance from the known global minimum to the location of
the best host nest with the lowest fitness value is evaluated. The average of the distance
difference for each loop from all the 30 trials is then measured. In addition, to authenticate the
statistical significance of the proposed ACSA, the two-tailed 𝑡-test is applied. The null
hypothesis is rejected at the confidence interval of 5% level, if the difference of the means of
both CSA and ACSA is statistically significant.

Algorithm of adaptive cuckoo search

Hybrid self-adaptive Cuckoo search algorithm
This section presents the proposed Hybrid self-adaptive Cuckoo search (HSA-CS), where the
original CS, as described in the previous, is modified by adding the following mechanisms:
 balancing the exploration strategies,
 self-adaptation of CS control parameters
 population reduction feature.
In the remainder of this paper, the proposed self-adaptive and hybrid mechanisms are described
thoroughly, accompanied by pseudo-code.3.1. Balancing the exploration
strategiesAccordingtoˇCrepinˇsek et al. in [36], the exploration/exploitation is achieved by
selection, mutation and crossover operators in EAs. A balancing between exploration and
exploitation is performed by a control parameter setting that is obviously a problem dependent.
The proposed HSA-CS algorithm implements three different strategies for exploring a search
space (also exploration strategies). Each of these strategies is controlled by their own control
parameters. Furthermore, a launching of the strategies is controlled by specific control
parameters. Consequently, both set of control parameters have great impact on the
exploration/exploitation components of the CS search process. According to the distance to how
farther trial solution can be generated from the parent solution, there are three exploration
strategies in the HSA-CS [35]:•the random long-distance exploration, the stochastic short-
distance exploration and the stochastic moderate-distance exploration.7
These strategies determine how different from the parent solution the generated trial solution
could to be expected after applying the specific exploration strategy. The first exploration
strategy presents the global random walk according to Eq. (4). Primarily, this strategy is devoted
for exploring new solutions and thus induced a rising of the population diversity. The second
strategy improves the current solution using the local random walk (RW) [46] according to Eq.
(2) and directs the search process to exploit the neighborhood of the already discovered solution.
The third exploration strategy ’randtobest/1/bin’ is borrowed from the DE algorithm [6] and
consists of two terms, i.e., the distance to the current best solution and the distance between two
random selected solutions. Intuitively, the first term has more exploitive, while the second more
explorative effect on the search process. Moreover, this strategy also introduces a crossover

operation. Typically, theSI-based algorithms employ only a mutation strategy, where all
elements of a trial solution are changed after using the operator. In addition to mutation ideas, a
crossover operator is applied to the trial solution in order to preserve some good elements of this
solution from being changed. While the mutations typically an unary operation, the crossover
operation demands an interaction between two or more population members and enables a flow
of information inside the population. This flow is controlled by the crossover rate (CR) that
limits the number of changed elements in the new solution. The proposed DE mutation strategy
is expressed as
u(t)i=x(t)i+Fi(x(t)best−x(t)i)+(x(t)r1−x(t)r2),
Where Fid notes a scaling factor regulating the magnitude e of the change, x (t) best is the
current best solution and, x (t) r1andx (t) r2denote the randomly selected solutions from a cuckoo
population. Introducing the crossover operator to the proposed DE mutation strategy inEq. (5)
Within the CS algorithm has a crucial impact on the performances, as turned out during the
experimental work. Mathematically, this crossover can be expressed as follows:
w(t)i,j={u(t)i,jrandj(0,1)≤CR∨ j=jrand,x(t)i,jotherwise,
Where CR∈ [0.0, 1.0] controls the fraction of parameters that are copied to the trial solution. The
condition j=j rand ensures that the trial vector differs from the original solutionx (t) in at least
one element. Let us notice that whenCR=1.0 the whole mutated vectoruiis copied to the trial
vectorvi.In this case, no crossover takes place. Finally, replacement of the randomly selected
solution is performed that can
x(t+1)k={w(t)iiff(w(t)i)≤f(x(t)k)∧ k=i,x(t)iotherwise,
Where=rand (0, NP) is a randomly selected integer number drawn from uniform distribution in
interval [0, NP).The main weakness of the majority of SI-based algorithms is fast convergence
toward a local optimum. Therefore, the biggest challenge for developers of these algorithms is
how to maintain the diversity of the population over the generations. The following scheme is
applied for balancing between two stochas-tic exploration search strategies that is controlled by a
balancing probability bin the HSA-CS algorithm
If {U (0, 1) ≤pb⇒moderatedistancestrategyotherwise⇒shortdistancestrategy

Algorithm of hybrid selfadaptive cuckoo search

Masterleaderslave cuckoo searchfor ANN optimizations
master-leader-slave for CS, which is based on the master-slave strategy with theaddition of
another unit called the leader. The leader unit does not involve the use of any optimization
procedure but it receives the best solutions found by the slaves if, after a certain number of
iterations, there is no improvement in the quality of the best solution. After receiving the best
solutions from the slaves the leader selects the best solution from among all thebest solutions
from the slaves. Then, the leader sends the information on the selected solutionto all the slaves to
guide them to follow this selected solution in the next LeÂvy flight. The masterreceives a copy
of the best solutions found by the slaves in each iteration and applies the CSoptimization
algorithm to the population of best solutions from the slaves. Then the overallbest solution is
updated and the master is zeroized. The cooperation between slaves, leaderand master is shown
schematically in Fig .
In Fig , the arrows labeled Solbest show that a copy of the best solution is transferred to the
master. While the arrows labeled f(best) denote that the information (quality) about the best
solution is sent to the leader. The arrows labeled f(b) illustrate that the information on the best
solution among all the best solutions from the slaves is sent to the slaves. The cooperation
between slaves and leader gives the algorithm a powerful exploration capability and provides a
high diversity of solutions in the population, while the support given by the master to the slaves

improves the ability of algorithm to achieve fast convergence. This proposed multi-
populationcooperative strategy is applied to improve the performance of the basic CS algorithm.
Thepseudocode of the proposed algorithm with modifications is shown in Fig
Algorithm of Masterleaderslave cuckoo searchfor ANN optimizations

Modified Adaptive Cuckoo Search(MACS)algorithm and formal description
for globaloptimization
Given enough computation, the CS will always find the optimum [5] but, as the search relies
entirely on random walks, a fast convergence cannot be guaranteed. Presented here for the first
time, two modifications to the method are made with the aim of increasing the convergence rate,
thus making the method more practical for a wider range of applications but without losing the
attractive features of the original method. The first modification is made to the size of the Levy
flight step size a. In CS, a is constant and the value a = 1is employed [11]. In the MCS, the value
of a decreases as the number of generations increases. This is done for the same reasons that the
inertia constant is reduced in thePSO [3], i.e. to encourage more localized searching as the
individuals, or the eggs, get closer to the solution. An initial value of the Lévy flight step size A
= 1 is chosen and, at each generation, a new Lévy flight step is calculated using
a ¼ A=
ffiffiffiffi
G
p
, where G is the generation number. This exploratory search is only performed on the fraction of
nests tobe abandoned. The second modification is to add information exchange between the eggs
in an attempt to speed up convergence to a minimum. In the CS, there is no information
exchange between individuals and, essentially, the searches are performed independently. In the
MCS, a fraction of the eggs with the best fitness are put into a group of top eggs. For each of the
top eggs, a second egg in this group is picked at random and a new egg is then generated on the
line connecting these two top eggs. The distance along this line at which the new egg is located
is calculated, using the inverse of the golden ratio
U ¼ ð1 þ
ffiffiffi 5
p
Þ=2, such that it is closer to the egg with the best fitness. In the case that both eggs have the
same fitness, the new egg is generated at the midpoint. Whilst developing the method a random
fraction was used in place of the golden ratio, it was found that the golden ratio showed

significantly greater performance than a random fraction. There is a possibility that, in this step,
the same egg is picked twice. In this case, a local Lévy flight searches performed from the
randomly picked nest with step size
a = A/G2. The steps involved in the modified cuckoo search are shown in detail in Algorithm 2.
There are two parameters, the fraction of nests to be abandoned and the fraction of nests to make
up the top nests, which need to be adjusted in the MCS. Through testing on benchmark
problems, it was found that setting the fraction of nests to be abandoned to 0.75 and the fraction
of nests placed in the top nests group to 0.25 yielded the best results over a variety of functions.
Algorithm of Modified Adaptive Cuckoo Search(MACS)algorithmand
formal descriptionfor globaloptimization

Discrete cuckoosearchalgorithm
Many applications of CS in different optimization problems have shown its promising
efficiency. For example, for both spring design and welded beam design problems, CS
obtained better solutions than existing solutions in literature (Yang and Deb, 2010).
However, many combinational optimization problems are discrete problems. In order to
apply CS into TSP, we propose a DCS algorithm based on basic CS and several good
strategies.
3.1 Definitions
Assume the number of the current city is C, the distance from city C to other cities is
A = [dC,1, dC,2, …, dC,m], where dC,C = +∞, m is the number of cities. And p = u/|v|–λ,
1 < λ ≤ 3. In different instance, the value of λ is different, p ∈ [0, +∞).
1 while p < 1, if dC,D = min(A), the nearest city D is the next city to be visited
2 while p ≥ 1, dist = dC,D × p, B = {j| dC,j ≤ dist, 1 ≤ j ≤ m} and dC,E = max(dC,j), j ∈ B
where dist denotes the flight distance of a cuckoo. Set B represents a set of cities that the
flight distance is effective, dC,E represents that city E is the farthest city from city C; city
E is a candidate city among the candidate set B.
In the process of solving TSP, the probability that each city is visited from city C is p,
p = [pc,1, pc,2, ···, pc,m]. The next visiting city is either the nearest city D or the farthest city
E.
3.2 Basic notion for solving TSP
Individual: A tour that a cuckoo visits all cities.
Population: All tours that n cuckoos visit all cities.
Flight: The move that cuckoo flight from a nest to another nest. The selective methods of
next nest refer to the description in Subsection 3.1.
Population initialization: Generate n nests (cities) randomly.
Fitness function: In this article, the length of a whole route is regarded as fitness.
Discard: The current nest is discarded according to the quality of individual, and new
nest is generated by Lévy flight.
Bulletin board: Record the optimal n tours obtained by cuckoos.

Initial circuit construction: A cuckoo visits all cities from its nest, and returns to its nest.
Each move use Flight operation. After the initial circuit (tour) is constructed, the initialcircuit
needs to improve by partial inversion operator. This partial inversion operator canbe described as
follows: assume current city is C, the next city is the nearest city D, ifcities C and D are non-
adjacent, then the segment between the farthest city E and thenearest city D (includes D) carry
out inversion operator, after that, the next city of Cperform same partial inversion operator. The
tour generated by the last city performs theinversion operator is added to bulletin board, and the
last city is regarded as new nest.New individual generation: Assume the current cuckoo nest is C;
the next visiting nest isD after Lévy flight. If the nests C and D are adjacent, then, a new
individual is generated,otherwise, the individual needs to perform the inversion operator
repeatedly until the totallength of the tour decreases. Figure 2 shows the flow chart of new
individual generation.Learning operator: The learning operator is similar to the crossover of IO.
The learningoperator acts on two individuals in the bulletin board at the same time. Assume there
aretwo individuals S1 and S2. If a substring in the S2 is better than S1, copy it to S1 and
change the bad of S1. If this substring is worse, copy corresponding substring from S1 to
S2 and mend the bad of S2. They study well from each other and change bad.
For example, assume that
S1 = (1, 2,3, 4,5,6,7,8),
S2 = (1,3,5,6,8, 4, 2,7),
where S1 and S2 are choose randomly from the Bulletin Board. Suppose the current city
C = 2, the operator search for the city S2, which is next to 2, so the substring is ‘2, 7’.Thus the
operator reverse the segment starts after city 2 and terminates after city 7 in theS1. Consequently,
a new individual is
S1′ = (1, 2,7,6,5,4,3,8),If the fitness of S1′ is less than S1 1 1 (FS′ < FS ), S1′ would be saved. If
not, it means thatthe substring ‘2, 7’ +’3, 8’ is worse than ‘2, 3’ + ‘7, 8’. So the corresponding
substring is‘7, 8’. Therefore the segment for inversion in the S2 starts after city ‘7’ and
terminatesafter city ‘8’. But ‘8’ is in front of ‘7’, the operator reverses the segment starts before
‘8’and ends before ‘7’. As a result, a new solution is producedS2′ = (1,3,5,6, 2,4,8,7).
If the fitness of S2′ is less than S2, S2′ would be saved. The learning operator make
C = 1, 2, ···, m in turn, For each value, the inversion caused by ‘study well’ and ‘change

bad’ has been applied several times in the process of executing.‘A’ operator: To optimise the
traversal path in the shape of an ‘A’ word, as shown inFigure 3. E.g., in path (1, 2, 3, 4, 5, 6, 7,
8), we chose A = 2, A visit next the city is B = 3,B the next city is C = 4. The cuckoo after a
flight, a city close to point B, suppose city forD = 7, D the next point to as E = 8. If AB + BC +
DE > AC + DB + BE, then the Bdeleted from the original position, add to between D and E.
After operation the path for(1, 2, 4, 5, 6, 7, 3, 8).There is also another kind of special situation,
city A and E coincidence. Assuming thecuckoo produced after a flight near the B city, then E =
2, and city A coincidence. InFigure 4, if BC + AD > AC + BD, then delete BC and AD, and add
AC and BD.3-opt: 3-opt analysis involves deleting three edges in a tour, reconnecting the tour in
allother possible ways, and then evaluating each reconnection method to find the optimum
one. This process is then repeated for a different set of three edges. An instance is used to
illustrate the 3-opt in Figure 4, three edges (a, b), (c, d) and (e, f) are deleted randomly in
tour S, the remaining segment are S1, S2 and S3. If there are two reconnections, and the
left reconnection is the best tour among all reconnections, this tour will replace the
original tour.

Algorithm of discrete cuckoo searchalgorithm
Step 1Initialize fitness and distance matrix.
Step 2 Generate initial population: n host nests xi(i = 1, 2,
···, n).
Step 3 Initial circuit is constructed by flight.
Step 4 The initial circuit is improved by partial inversion; the
best tour is saved intobulletin board.
Step 5 When termination criterion is met, go to Step 14.
Step 6 Generate new individual by Lévy flight and evaluate its
quality (fitness) Fi.
Step 7 Update the optimal tour on the bulletin board by learning
operator.
Step 8 Update the optimal tour on the bulletin board by ‘A’
operator.
Step 9 Update the optimal tour on the bulletin board by 3-opt.
Step 10 If the new tour is better than original tour, the
original tour is replaced by thisnew tour on bulletin board.
Step 11 The worse nest is discarded with probability pα ∈ (0,
1), and then a new nest isgenerated by Lévy flight.
Step 12 Rank the tours on bulletin board and find out the
current optimal tour.
Step 13 The iteration variable is updated, and goes to Step 5.
Step 14 Output results.

DATA SET IMPLEMENTATION
(BEST FITNESS VALUES) AND
COMPARSIONSOF ALGORITHMS
AND FUNCTIONS

Now, after discussing algorithms we will find the data about their related variant of
cuckoo search.
To find data, we use the variants of cuckoo search algorithms with different
benchmark functions and the data will be in the form of BEST FITNESS VALUE
(OBJECTIVE VALUE) for each value. We will select only MEAN column
from the data set.
The data that we used …

In this we just used the mean values.

Now after collecting all data, we put the data in an excelsheet

After collecting data, now we give the Rankings to the above data

After ranking, now find the average of the ranking

After find average, now we find the ranking of average
Now the step move towards to find the variance of given ranking
To find the variance of given ranking
As we see that 1 has lowest ranking among all of average series now we take just 2
values first lowest and second lowest. Here we have the first lower minimum value
is 1 and second minimum value is 2. From the usage of these rankings, we find out
the variance by using the ranking columns of both values. To find variance, we use
the formula =var()with the help of Microsoft excel.

After finding variance, the smallest value of variance is find from ranking 1so we
will consider it. After that, we will find the max rank and min rank values.

To find the max rank,
After finding variance we find max rank by using the value of variance in such a
way that it will be added in value of ranking, as we know the smallest value of
variance is 2.1098901which is find out through the ranking 1so we will add it
into it.
To find the min rank,
After finding variance we find min rank by using the value of variance in such a
way that it will be subtracted from the value of average, as we know the smallest
value of variance is 2.1098901which is find out through the ranking 1and the
average of ranking 1 is 2.482571sowe will subtract from it.

Now after collecting data and apply all functions, put the data in Medcalc
statisticalsoftware to compare the multiple algorithms.
1) If we use the algorithms name in columns then we will get the output of the
following

Then select “Anova” and apply Friedman test

Add the columns in the variable section and set significance levelto 0.01

Here the multiple comparisons in which we see that after applying testing we see
the value P which is less than <0.000001

the value P which is less than <0.000001 and in multiple comparisonsections you
can check the comparison among them.

Here is the graph of each column
Overall summary of all values

2) If we use the function name in columns then we will get the output of the
following

Then select “Anova” and apply Friedman test

the value P which is equal to 0.000001 and in multiple comparison sections you
can check the comparison among them.

Here is the graph of set of five functions.
Overall summary of graph

Final report aaa 2

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