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# RELIABILITY OPTIMIZATION WITH NATURE BASED ALGORITHM

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RELIABILITY OPTIMIZATION WITH NATURE
BASED ALGORITHM
Dr. Vikrant Aggarwal1, Dr. Rakesh Kumar2, Dr. Harmeet Singh3, Dr. Yogesh Kumar4
1General Manager, SV Woods, Hoshiarpur, Punjab, India
2Principal, Department of Mechanical Engineering, Ludhiana Group of Colleges, Chokimann, Ludhiana,
Punjab, India
3HOD and Associate Professor, Department of Mechanical Engineering, Rayat Institute of Engineering &
Information Technology, Ropar, Punjab, India
4Principal, Department of Electronics Engineering, CT Group of Institutions Jalandhar, Punjab, India
E-mail: vikrant0805@gmail.com1, lgcengg.principal@gmail.com2, harmeet_pabla@yahoo.com3
Received: 14 March 2020 Revised and Accepted: 8 July 2020
ABSTRACT: Reliability and maintenance engineering is a powerful tool which enable the industries to find
ways of costs savings and operational improvement opportunities. In this paper, a continuous production system
is considered for evaluating and analysing the system‟s availability. The availability of the production system is
evaluated by using Markov Modelling. Differential equations are derived and solved by Laplace transform to
attain state probabilities. Further, the availability optimization has been done using four nature-based
algorithms: Moth Flame Optimization, Dragonfly Algorithm Optimization, Ant Lion Algorithm and Whale
Optimisation Algorithm. The obtained results of optimization has been compared and with the present
availability of the system.
KEY WORDS: Optimization, reliability, WOA, MFO, ALO, DAO .
I. INTRODUCTION
The purpose of optimization is to find the best combination of input variables to maximize or minimize the
objective function for a particular problem. In everyday life, engineers, researchers, scientist and statisticians
have to take various managerial or technological decisions [1]. For solving any optimization problem, various
steps need to be followed. Firstly, the optimization problem is classified as continuous or discrete depending on
the nature of the input variables. Secondly, the various constraints applied to the objective function has to be
identified. The constraints divide the optimization problem into constrained or unconstrained. In the next step
the problem, the objectives of the optimization problem are investigated and classified as a multi-objective or
single-objective problem. Finally, an algorithm is decided in consideration with the nature of constraints,
parameters and the objective function.
Meta heuristic algorithms are often used for attaining optimization [2]. Broadly, these can be categorised as
evolution based techniques, physics-based techniques and swarm-based techniques. Evolution-based
optimization techniques are based on the laws of natural evolution. The most widely used evolution based
optimization technique is a Genetic algorithm, based on Darwin‟s theory originally proposed by Holland
[3].The various other popular evolution-inspired techniques are Genetic programming, Neuro-evolution,
learning classifier system and Biogeography Optimizer. Physics-based optimisation techniques replicate the
physical laws of the universe. Swarm-based techniques replicate the social behaviour of animals. Commonly
used algorithm is Particle Swarm Optimization [4].The algorithm mimics the social behaviour of bird flocking.
Another popular swarm-based met heuristic algorithm namely, Cuckoo Search proposed by Yang and Deb [5].
It mimics the cuckoo‟s behaviour of laying eggs in the nests of other birds. In recent years, a new optimization
technique called dragonfly algorithm (DA) was developed by Mirjalili [6]. Mirjalili [7]also proposed Moth-
Flame Optimization (MFO) algorithm inspired from the navigation method of moths. In the year 2016, Mirjalili
and Lewis [8] proposed another nature-based algorithm known as Whale Optimization Algorithm. Table 1
represents the various swarm-based optimization techniques:
Table 1. Various swarm-based optimization techniques.
Inspiration
Algorithm
Year of
proposal
Bird Flock
Particle Swarm
1995
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optimization [4]
Fish
Swarm
Artificial fish swarm
algorithm [9]
2003
Termite
colony
Termite algorithm [10]
2005
Ant Colony
Ant colony
optimization [11]
2006
Parasitic
Wasp
Wasp Swarm
Algorithm [12]
2007
Cuckoo
Cuckoo search [5]
2009
Bat Herd
Bat-inspired algorithm
[13]
2010
Group of
animals
Hunting search [14]
2010
Bird
mating
Bird mating Optimizer
[15]
2012
Dolphin
Dolphin echolocation
[16]
2013
Moth
Flame
Moth Flame
Optimization [7]
2015
Dragonfly
Dragonfly Algorithm
[6]
2016
Humpback
Whale
Whale Optimisation
Algorithm [8]
2016
Nature Based algorithms proved their effectiveness in every field. The wide application of algorithms really
helps in maximizing the objective function under various conditions. In the present paper, the various
techniques are applied to optimize the system‟s availability. The algorithms though improve the availability, but
the present paper focus on finding the best algorithm among the various algorithms.
II. OVERVIEW OF OPTIMIZATION METHODS
The various optimization methods used in present analysis has been discussed below.
Moth Flame Optimization
Moths fly in the night by preserving a particular angle with the moon. This will make the moths possible, to fly
long distances in a straight line [17]. Moth-Flame Optimization (MFO) [7] is inspired by this unique navigation
method of moths. However, moths are stuck in a spiral path around artificial lights. As the light source is very
near, moths try to retain a similar angle with the light source as they maintain with the moon. [18]This result in
the deadly spiral fly path. The moths can fly in 1-Dimesion, 2-Dimension, 3-Dimension or hyper dimensional
space. The solutions in the algorithm are both flames and moths. The difference between them is the way these
been updated in each iteration. A logarithmic spiral has been chosen as the update mechanism of moths.
Mathematically it can be represented as: S(Mk,Fk)=Dkebt cos(2πt) + Fj (1)
Where “Dk” represents the distance of kth moth for jth flame, “b” is a shape constant and “t” is a random number.
The variable “Dk”can be calculated as, (2)
Where “Mk” indicates the kth moth and Fj indicates the jth flame.
The position of moths with respect to “n” different locations can be updated by adopting an adaptive
mechanism. 

Where
“l”, “N” and “T” represents the maximum number of iteration, flames and iterations respectively. In the final
stages of iteration, the moths update their positions with respect to the best flame.
Ant Lion Optimization
The Ant Lion Optimizer (ALO) [19] is a meta-heuristic technique based on the interaction of ant lions and ants
in nature. Ant lion lives in two phases of larvae and adult. Their hunt mechanism is unique when they are
larvae. Ant lions make a small cone shape pits to trap ants. They sit under the pit and wait for ants to fell. After
consuming the prey‟s flesh, ant lions throw the remains outside the pit and alter the pit for the next hunt. It has
been observed that ant lions tend to dig a bigger pit when they are hungry.
The random walk in the ALO is as under:-
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X(t)=[0,cs(2r(t1)-1), cs (2r(t2)-1)……… cs (2r(tn)-1)] (4)
 
 (5)
Where,
cs” is the cumulative sum, “t” represents the step of random walk and “r(t)” is a function
The position in a random walk is updated by the equation (5) 
 (6)
Where,
ai and bi” represents the min. and max. of the random walk of the ith variable.
The effect of ant lions is exhibited as:  (7)
Where,
ct is the minimum value at the tth iteration
For Building traps, roulette wheel technique is used. The ant lions are selected based on their fitness. In the
algorithm, catching of target occurs when ants dump inside the pit. An ant lion is needed to change its position
tomaximise its chance of grabbing prey. The following equation shows the behaviour:
  (8)
where “t” represents the current iteration.
The concept of Elitism lets the algorithm to maintain the best solution. In ALO, the best ant lion obtained in
each iteration as elite. The movement of other ants is affected by the elite ant. During iteration, every ant walks
around a selected ant lion as follows: 

Where“
and
represents the random walk around the ant lion and random walk around the elite at tth
iteration respectively. indicates the position iteration.
Dragonfly Algorithm (DA)
The algorithm [6]instigates from behaviour of dragonfly insect. The two swarming behaviours are exploration
and exploitation. In exploration, Dragonflies make sub swarms and fly over various areas. Whereas in
exploitation phase, dragonflies fly in larger swarms in unidirection. Reynold [20] proposed various principles of
swarming: separation, alignment, cohesion, attraction to a food source, a distraction from enemies. These
perceptions allow to simulate the behaviour of dragonflies. The algorithm has two vectors: step vector and
position vector. The main equations for these two vectors are:
  (10)
Where,“s” and “a” is separation weight and alignment weight respectively, and “A” represents the
separation and alignment of the ith individual respectively, “c” represents the cohesion weight, , “e” , “w”
and “f” represents the cohesion , enemy factor, inertia weight and food factor respectively,and
represents the food source and position of enemy of ith individual and “t” represents the iteration counter.

 






Where “Z”, “” and “ represents the positions of the current dragonfly, food source and enemy, N
represents the number of neighbouring dragonfly and indicates the position of  neighbouring solution.The
position of dragonflies are updated with the following equation:   (16)
Whale Optimisation Algorithm
WOA is based on the hunting method of humpback whales [21]. Humpback whales usually hunt school small
fishes, near the water surface. they have spindle cells in some parts of their brain. These cells control the
judgement, social and emotion behaviour in humans. Due to the presence of spindle cells in the brain, the whale
can judge the things, communicate, learn and become emotional. Sometimes whales develop their own dialect
for communication. The whales can identify the position of prey and circumscribe them. WOA algorithm
considers the current best candidate to be the target prey. Mathematically, the behaviour can be shown as:


 (17)
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
(18)
Where, “t” represents the current iteration,  are coefficient vectors, and
is the position vector of
prey and whale respectively.
The bubble net behaviour of the humpback whale is modelled by calculating the distance between whale and
the prey . Considering the position of the whale and prey, an equation corresponds to helix movement of the
whale is:
 (19)
Where,

(20)
“b”and “t” represents the constant for defining the shape of the spiral and random number respectively.








(23)
Fig. 1 Block diagram of Rubber Tube Extraction system
Rubber Tube Extraction System
Ramson Cycle Pvt. Limited is a medium size rubber cycle tube manufacturing industry, situated in Ludhiana
(Punjab), India. The organization is manufacturing various types of rubber cycle tubes in both moulded and
jointed form according to the demand. The breakdowns and failures are quite frequent in the plant affecting the
availability of the system. The company employs both Preventive Maintenance (PM) and Corrective
Maintenance (CM). The PM is performed when the system components come to pending-to-failed state and CM
is performed in reaching the component to failed state respectively. For manufacturing the Rubber cycle tubes,
initially an appropriate proportionate of raw rubber, Clays, Minerals, activators and oil are added for the
Strainer
Refinement Mill
Extruder-I
Extruder-II
Extruder-III
Extruder-IV
Extruder-V
Cutter -I
Cutter -II
Cutter -III
Cutter -IV
Cutter-V
Rubber Sheet Making System
Vulcanizer
Tube Welding
Storage
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preparation of the master batch of rubber. The batch is mixed continuously for 8 minutes in Kneaders. Each
Kneader has a capacity of 92.325 kg. After proper mixing, the batch goes to Mixing mill. Here, rubber sheets of
approximately 4 feet x 3 feet x 3.5 inches are formed. After proper ageing, sheets are warmed for 15 minutes in
a Warming Mill to further enhance the properties of rubber. The rubber sheets are transferred to Strainer, to
remove foreign particles of metal or dust. These sheets are then sent to Refinement Mill where the powdered
accelerators are impinged on sheets. The rubber tubes will be formed with the help of Extruders and cut with
Cutters. These extruded tubes are treated in Vulcanizers for further enhancing the rubber properties. The
vulcanized tubes are welded according to the requirement. The welded tubes are packed and stored in the
storage. In the present paper, we will consider the Rubber Tube Extraction Subsystem for analysis and
optimization. The block diagram representing the working of the Rubber Tube Extraction Subsystem is shown
in Fig.1. The subsystem comprises of five different components.
1. Component W1: It consists of one Strainer. If it fails, the complete failure of the subsystem takes place.
2. Component W2: It consists of one Refinement Mill. If it fails, the complete failure of the subsystem takes
place.
3. Component W3: It consists of five parallel units (W31, W32, W33, W34& W35), each unit comprising of one
Extruder and one Cutter in series. This component can work with three parallel units in the reduced
capacity. If more than three units fail, the subsystem will be in the failed state.
4. Component W4: It consists of one Vulcanizer. As Vulcanizer has a negligible failure rate, it will not affect
the subsystem's availability.
5. Component W5: It consists of one Tube Welding unit. As Tube Welding has a negligible failure rate, it will
not affect the subsystem's availability
The components of Rubber Tube Extraction subsystem are subjected to various failures, which cause the
decrease in the subsystem‟s availability. Table 2 represents the common failures that occur in various
components of the Rubber Tube Extraction subsystem.
The various notations, assumptions & formulation of the subsystem used in modelling the Rubber Tube
Extraction system are as under
Superscript “o” : represents component is operative.
Superscript “r” : represents component is under repair.
Superscript “g” : represents component is good but not operative.
Superscript “qr” : represents component is in queue for repair.
Superscript “m” : represents component is under PM.
Superscript “qm” : represents component is in queue for preventive maintenance.
s : Laplace transform variable
λ1, λ2 : represents failure rates of components W1& W2 respectively
λ3 : represents failure rate of component of W3
: represents failure rate of units of W3. (i=31, 32, 33, 34 &35)
μ1, μ2 : represents repair rates of components W1& W2 respectively
μ3 : represents repair rate of the complete component W3 initiated on failure of four or more parallel
units of component W3
: represents repair rate of units of W3 (i=31, 32, 33, 34 & 35)
β1, β2 : represents transition rate leading the W1& W2 respectively to go for preventive
maintenance schedule
θ1, θ2 : preventive maintenance rate of components W1& W2 respectively

 : represents the working status of the component W3, which consists of five identical
units W31, W32, W33, W34& W35 working in parallel. The subscript „m‟ represents the number of working units
of W3 while „n‟ represents the number of failed units of W3. Pair
designates the working status of „m‟
working units, while pair
represents the repair status of „n‟ failed units and this pair can further be expanded
as 
 if „n‟ becomes more than one.
Table 2 Common failures and their remedies of various components of the Rubber Tube Extraction
system
Component
Common Failures/problems
Remedies
Strainers
V belt breakage
Replace
Worm gear breakage
Replace
grill breakage
Replace
Refinement
mill
V belt breakage
Replace
Worm gear breakage
Replace
Electric panel
Repair/replace
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Extruders
V belt breakage
Replace
Worm gear breakage
Replace
Cutter
Electric panel breakdown
Replace
Assumptions
1) At any given time, the component is in operating, reduced or in the failed state.
2) Failure, repair, preventive maintenance & transition rates are constant.
3) A repaired component is as good as new.
4) Repair facilities are always available.
5) The plant employs a single maintenance team to handle both preventive maintenance & corrective
maintenance.
6) At a time, preventive maintenance can be performed on one component.
7) The component is as good as new after preventive maintenance.
8) If any component of the subsystem is in repair, preventive maintenance of the other component will
not be initiated.
9) The subsystem will work at 80 % efficiency in with four units of W3, 60 % efficiency when working
with three units of W3 and 40 % efficiency with two units of W3 in working state.
Formulation of the equations
The availability of a Rubber Tube Extraction Subsystem is determined by the mathematical formulation for a
transient state of the model using first order differential-difference equations. Fig. 2 represents the state
transition diagram of the Rubber Tube Extraction Subsystem.



(24)


(25)



(26)



(27)



(28)




 (29)



(30)

 (31)

 (32)

 (33)

 (34)

 for a=0,1………5 (35)
Where,










Solving recursively the above equations after taking Laplace transform of equations (24) (35), the following
Laplace transforms of state probabilities are obtained:
, For n= 1 to 35 (36)
Where,








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













  
   
 , 
Taking Laplace transform of equation (23)
 (37)
Where,



 
Full capacity availability function AFC(s) for the Rubber Tube extraction Subsystem is given as

Reduced capacity availability function ARC6(s) for the Rubber Tube extraction Subsystem.is given as

Inversion of AFC(s) and ARC(s) gives the availability function AFC(t) and ARC(t) respectively.
Steady state availability of Rubber Tube Extraction system
The steady state behaviour of the system can be analyzed by setting t→ ∞ and  0, in the equations (24)
to (35).


  
 

  

   

    


   

    
 
  
 
  
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Fig. 2 State transition diagram of Rubber Tube Extraction system
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 
Where,




















   
   
From equation (40), we get


 
Using normalizing condition,

 


 
The full availability function AFC (t) for Rubber Tube Extraction Subsystem is given as,
 
The reduced availability function ARC (t) for Rubber Tube Extraction Subsystem is given as,
     
    (54)
For calculating & analysis of the availability of the subsystems, the feasible values and range of the failure,
repair, transition & preventive maintenance rates has been selected considering the following:-
1. Data of the past two years from the maintenance reports, log books, history cards, daily reports etc.
2. Detailed discussions with skilled plant personnel about the component performance.
3. By observing & deep study of the system‟s working over a period of one year.
Table 3 represents failure, repair, transition & PM rates of various components of the subsystem. Substituting
the values in equation (53) for AFC& in equation (54) for ARC we get:
AFC=0.8454
ARC=0.0395
The Rubber Tube Extraction Subsystem is running at full capacity or reduced capacity. While running at
reduced capacity, the availability reduces to 80% when working with four units of W3, 60% when working with
three similar units and 40% when working with two units of W3. Therefore, the total availability (ATC) of the
subsystem can be computed as:
ATC = AFC6+0.8* (P5 + P15 + P30) +0.6*(P10 + P25) + 0.4*P20
= 0.8454+0.8* (0.037442) +0.6*(0.00197275) + 0.4*0.00007437
ATC = 0.8766
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Table 3 Failure, repair, transition and PM rates of the components of Rubber Tube Extraction system
Component
Failure
Rate
Repair
Rate
Transition
Rate
PM
rate
Strainers
0.01
0.2
0.004
0.4
Refinement
Mill
0.012
0.2
0.005
0.5
Extruders &
Cutters
0.012
0.35
-
-
III. COMPUTATION RESULTS, ANALYSIS AND DISCUSSION
Various algorithms were coded in C++ language for optimising the availability. The task of optimizing was
performed in 100 runs, with each run iterated until the best solution was found. The local maxima were found
by using the maximum and minimum value of the input variables. The values are chosen such that the extreme
values are ± 25% of the mean values. In the present work, the newly developed optimization techniques: Moth
Flame Optimization, Dragonfly Algorithm Optimization, Ant Lion Algorithm and Whale Optimisation
Algorithm have been used for optimizing the total availability of the subsystem. The objective function must
approach to maximum value within the specified range of input variables. The lower and upper bound of the
input variables (failure and repair rates) used in the availability optimization has been shown in table 4.
Table 4. Minimum and Maximum values of input variables
Paramete
r
λ1
λ2
λ3
μ1
μ2
μ3
Minimu
m
0.007
5
0.009
0.009
0.15
0.15
0.262
5
Maximu
m
0.012
5
0.015
0.015
0.25
0.25
0.437
5
In reliable systems, a small increase in reliability is difficult to be achieved. Increasing reliability will only be
possible with the additional costs. The cost may be incurred due to additional skilled labour or specialised
maintenance engineer, upgradation in machinery or equipment, process or design modifications, additional
inventory of spare parts and lubricants etc. The introduction of redundant components may be another option to
increase the availability of the system. The results of optimization of availability with various nature based
algorithms are shown in Table 5. The best solutions obtained are reported corresponding to optimization
technique. The desired total availability (ADTC)of the subsystem, is best obtained by WOA. The ADTC of 91.63%
can be achieved with λ1=0.0075, λ2=0.009, λ3=0.0091, μ1=0.2498, μ2=0.25 and μ3=0.4375.
Table 5 Best result for the ATC with various algorithms
Algorithm
λ1
λ2
λ3
μ1
μ2
μ3
MFO
89.98
0.0077
0.0091
0.0093
0.2487
0.249
0.4369
DAO
90.52
0.0076
0.0091
0.0094
0.2492
0.2492
0.4375
ALO
90.24
0.0075
0.0091
0.0093
0.2495
0.2493
0.4371
WOA
91.63
0.0075
0.009
0.0091
0.2498
0.25
0.4375
The effectiveness of the algorithm in optimizing the availability of the subsystem is measured in terms of
Maximum Improvement Percentage (MI %). MI% can be defined as the percentage improvement in the
availability found by the algorithm to the present total availability. Higher MI% signifies the higher
improvement in the availability of the subsystem, whereas lower MI% indicates the poor improvement in the
subsystem‟s availability.  
 
Where,
= best system availability obtained corresponding to the algorithm
= Present availability of the system as described in section 2.3
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Table 6 & fig. 3 demonstrates the MI% in availability with various nature based algorithms. Although all the
algorithms improve the availability of the subsystem. But the maximum M I % in the total availability of 4.53%
is obtained with Whale Optimisation Algorithm, whereas the minimum M I % of 2.65 in the total availability is
obtained with Moth Flame Optimization.
Table 6. M I % in availability with various algorithms
Algorithm
Maximum
Improvement %
Moth Flame
Optimization
2.65
Dragonfly Algorithm
3.26
Ant Lion Algorithm
2.94
Whale Optimisation
Algorithm
4.53
Fig. 3 Variation in the availability improvement with various algorithms
It was observed that WOA‟s search agents tend to search likely regions of design space and find the best one.
Search agents change rapidly in the initial stages of the optimization and then slowly converge. Such a
behaviour of an algorithm can guarantee that a population- based algorithm ultimately convergences to a point
in a space [47]. Convergence curves of MFO, DAO, ALO and WOA are compared in Fig. 4 for problem. It can
be seen that WOA is enough competitive with other algorithms. As observed in this ﬁgure, the WOA algorithm
shows two different convergence behaviours while optimizing the problem. Firstly, the convergence of the
WOA likely to be accelerated as iteration increases. This is due to the adaptive mechanism of WOA that helps it
to look for the promising regions in the initial steps of iteration and more rapidly converge towards the
optimum after passing almost half of the iterations. The second behaviour is the convergence towards the
optimum only in ﬁnal iterations. This is probably due to the failure of WOA in ﬁnding a good solution for
exploitation in the initial steps of iteration when avoiding local optima. WOA keep looking for the search space
to ﬁnd good solutions.
IV. CONCLUSION
This paper presented a relative investigation of different algorithms to optimize the availability of a production
system. Various four algorithms are used to find the local maxima of the availability of the system under ideal
and faulty PM. any production system can never be free from failures. The present work optimizes the
production system by considering ± 25% of the present values of failure, repair, and transition and PM rates.
The local maxima have been found. By making necessary changes and adjusting the input parameters, the
system can be optimized for enhanced availability. The algorithms specify the values of parameters conforming
to the availability .Besides this, a recently developed algorithm; WOA finds its worth for reliability
optimization. The application of these algorithms may prove to be favourable in enhancing the availability in
the given constraints.
0
1
2
3
4
5
MFO DAO ALO WOA
Maximum Improvement %
Algorithm
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.
Fig. 4
Convergence curves of
MFO, DAO, ALO and WOA
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
020 40 60 80 100
Availability
Number of Iterations
MFO DAO
ALO WOA
ISSN- 2394-5125
VOL 7, ISSUE 19, 2020
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Authors Profile
Dr. Vikrant Aggarwal is a working as General Manager in SV Woods, Hoshiarpur.
He has done P.Hd in mechanical engineering from IKGPTU, Kapurthala. He has
more than 15 years of teaching experience and 2 years of industrial experience. His
Research interests are Reliability Engineering and Industrial Engineering. He has
published 8 national and international research papers.
Dr. Rakesh Kumar is working as Principal in Ludhiana Group of Colleges,
Chokimann (Ludhiana).He has done Ph.D in Mechanical Engineering Department
from IKGPTU, Kapurthala. He has more than 15 years of teaching experience. His
research interest is in Manufacturing Engineering. He has published many national
and international research papers.
Dr. Harmeet Singh is working as HOD in Department of Mechanical Engineering,
Rayat Institute of Engineering & Information Technology, Ropar. He has done Ph.D in
Mechanical Engineering Department from IKGPTU, Kapurthala. He has more than 12
years of teaching experience and 2 years of Industrial Experience. His research
interest is in Manufacturing Engineering. He has published many national and
international research papers.
Dr. Yogesh Kumar is working as Principal in CT Group of Institutions, Jalanhar.He
has done Ph.D in Electronics Engineering Department from Desh Bhagat University,
Mandi Gobindgarh. He has more than 20 years of teaching experience. His research
interest is in Nano Electronics. He has published many national and international
research paper
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Article
This paper proposes a novel nature-inspired algorithm called Ant Lion Optimizer (ALO). The ALO algorithm mimics the hunting mechanism of antlions in nature. Five main steps of hunting prey such as the random walk of ants, building traps, entrapment of ants in traps, catching preys, and re-building traps are implemented. The proposed algorithm is benchmarked in three phases. Firstly, a set of 19 mathematical functions is employed to test different characteristics of ALO. Secondly, three classical engineering problems (three-bar truss design, cantilever beam design, and gear train design) are solved by ALO. Finally, the shapes of two ship propellers are optimized by ALO as challenging constrained real problems. In the first two test phases, the ALO algorithm is compared with a variety of algorithms in the literature. The results of the test functions prove that the proposed algorithm is able to provide very competitive results in terms of improved exploration, local optima avoidance, exploitation, and convergence. The ALO algorithm also finds superior optimal designs for the majority of classical engineering problems employed, showing that this algorithm has merits in solving constrained problems with diverse search spaces. The optimal shapes obtained for the ship propellers demonstrate the applicability of the proposed algorithm in solving real problems with unknown search spaces as well. Note that the source codes of the proposed ALO algorithm are publicly available at http://www.alimirjalili.com/ALO.html.
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Distinct behavioral differences were noted from aerial observations of four species of baleen whales (Eubalaena glacialis, right whale; Balaenoptera borealis, sei whale; Megaptera novaeangliae, humpback whale; Balaenoptera physalus, finback whale) feeding together on 30 April and 1 May 1975. The right and sei whales fed together on patches of plankton. Right whales fed steadily with mouths open in the densest areas, while the sei whale followed a faster but more erratic path through the patches, alternately opening and slowly closing its mouth with slight throat distension at each closing. Humpback and finback whales fed together on dense schools of fish associated with the patches of plankton. The humpback fed by rushing, generally from below the schools of fish, while finback feeding was by more horizontal passes sometimes characterized by sharp turns and rolls within the fish schools and often with enormous throat distension.
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Many design problems in engineering are typically multiobjective, under complex nonlinear constraints. The algorithms needed to solve multiobjective problems can be significantly different from the methods for single objective optimization. Computing effort and the number of function evaluations may often increase significantly for multiobjective problems. Metaheuristic algorithms start to show their advantages in dealing with multiobjective optimization. In this paper, we formulate a new cuckoo search for multiobjective optimization. We validate it against a set of multiobjective test functions, and then apply it to solve structural design problems such as beam design and disc brake design. In addition, we also analyze the main characteristics of the algorithm and their implications.