Circuits and Systems, 2016, 7, 735-747
Published Online May 2016 in SciRes.
How to cite this paper: Rajeswari, D. and Senthilkumar, V.J. (2016) Minimizing Time in Scheduling of Independent Tasks
Using Distance-Based Pareto Genetic Algorithm Based on MapReduce Model. Circuits and Systems, 7, 735-747.
Minimizing Time in Scheduling of
Independent Tasks Using Distance-Based
Pareto Genetic Algorithm Based on
MapReduce Model
Devarajan Rajeswari*, Veerabadran Jawahar Senthilkumar
Department of Electronics and Communication Engineering, College of Engineering, Anna University,
Chennai, India
Received 21 March 2016; accepted 9 May 2016; published 12 May 2016
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativ ecommon icenses/by /4.0/
Distributed Systems (DS) have a collection of heterogeneous computing resources to process user
tasks. Task scheduling in DS has become prime research case, not only due of finding an optimal
schedule, but also because of the time taken to find the optimal schedule. The users of Ds services
are more attentive about time to complete their task. Several algorithms are implemented to find
the optimal schedule. Evolutionary kind of algorithms is one of the best, but the time taken to find
the optimal schedule is more. This paper presents a distance-based Pareto genetic algorithm (DPGA)
with the Map Reduce model for scheduling independent tasks in a DS environment. In DS, most of
the task scheduling problem is formulated as multi-objective optimization problem. This paper
aims to develop the optimal schedules by minimizing makespan and flow time simultaneously.
The algorithm is tested on a set of benchmark instances. MapReduce model is used to parallelize
the execution of DPGA automatically. Experimental results show that DPGA with MapReduce mod-
el achieves a reduction in makespan, mean flow time and execution time by 12%, 14% and 13%
than non-dominated sorting genetic algorithm (NSGA-II) with MapReduce model is also imple-
mented in this paper.
Distributed Systems, Multi-O bjective, MapReduce, Opti mization , DPGA, NSGA-II
Corresponding author.
D. Rajeswari, V. J. Senthilkumar
1. Introduction
The Computationa l power of individ ual system is not suf ficient for solving widel y used complex co mputatio nal
tasks like high energy physics, earth science, etc. In order to solve complex jobs, high performance parallel and
distributed systems are developed with a number of processors. As the computing nodes are heterogeneous in a
multiprocessor environment, execution time of tasks varies on each processor. Scheduling of tasks is a key issue,
to achieve the high usability of supercomputing capacity of distributed computing environment [1]. To ensure
efficient utilization of resources, suitable scheduling algorithms are used to assign the tasks to the available pro-
cessors efficiently.
For distributed computing environment, static scheduling can be used due to geographically distributed com-
puting resources with various ownerships, access policy and different constraints. Schedulers are developed us-
ing complex arithmetic techniques that use the available values of application and environment. So, heuristic
methods are the best approach to find the optimal schedule in the DS enviro nment [2]. The most important crite-
ria used to analyze the efficiency of scheduling techniques are makespan and flowtime. Time taken to complete
the last task is Makespan [3] and the sum of completion time of all tasks in a schedule is flowtime. The sche-
dule, which optimizes the makespan and flowtime is called as optimum schedule [4]. To minimize makespan,
the Longest Job to be scheduled to Fastest Resource (LJFR) and for minimizing flowtime, the Shortest Job to be
scheduled to Fastest Resource (LJFR) [2]. Flowtime minimization makes the makespan maximization. This
leads the pro blem as multiple o bjective.
Genetic Algorithm (GA) is a search and population-based model [5]. This has b een exte nsivel y used i n vari-
ous problem domains. GA has the capability to search various regions of the solution space and note a diverse
set of solutions for the distributed computing problem. GA uses genetic operators to improve the structure of
good solutions in various objective spaces. These characteristics of GA used to find the best optimal schedule
for multi-objective problem in distributed systems. The distance-based Pareto genetic algorithm (DPGA) of
Osyczka [6] is used in this paper. DPGA uses a distance computation and dominance test procedure and elitist
method of combining parent populatio n with the of fspring po pulation for t he next iteratio n. Set of most d ifficult
static benchmark instances of Braun et al. [1] is used to analyze the performance of NSGA-II. The degree of
task and resource heterogeneity can be captured by these instances. Hadoop MapReduce is a framework for dis-
tributed large scale data processing on computer clusters. MapReduce is used to automatically parallelize the
data processing on clusters in a reliable and fault-tolerant manne r [7].
MapReduce programming model is used to implement the multi-objective DP GA to find the o pti mal schedule
with minimum time in the distributed computing environment. The DPGA with MapReduce model suits for dis-
tributed computing environments with various co mputing resources and scheduling is done by considering ma-
kespan and flowtime minimization. The DPGA with MapReduce model generates better solutions in minimal
time than N SGA-II with MapReduce model.
The remainder of the paper is structured as follows. Section 2 discusses t he literature revie w. Multi-objective
optimization introduction is presented in Section 3. Section 4 determines for identifying the variety of distri-
buted c o mput i n g e nvir o n me nt in t he s i mula t io n p r oc e ss. A Sc hed ul in g method usi n g NSGA-II with MapReduce
and DPGA with MapReduce is available in Section 5. Simulation results are shown in Section 6. Finally, Sec-
tion 7 concludes and discu s s es the future work.
2. Related Work
Optimal scheduling of independent tasks to available resources in DS is an NP-complete problem and it depends
on var ious heur istics a nd meta -heuristic algorithms. A fe w well kno wn heur istic metho ds ar e min-max [8], suf-
frage [9], min-min, max -min [10] and LJFR-SJFR [11]. These above heuristic methods are more time consum-
ing process. In recent, several meta-heuristic methods are developed to solve complex computational problems.
The most popular methods are GA [6], particle swarm optimizatio n (PSO) [12], ant colony opti mization (ACO)
[13] and simulate d a nneal in g ( SA) [14]. The description of eleven heur isti cs a nd co mpar i son o n the va rio us d is-
tributed environment was done by Braun et al. [1] and illustrates the effectiveness of GA with others. All the
above meta-heuristic methods considered single objective and aimed to minimize the makespan.
There are some methods considered multiple objectives, while scheduling tasks in distributed environments.
Izakian et al. [15] compares five heuristics depends on the machine, and task characteristics for minimizing both
makespan and flowtime, but calculated separately. Several nature inspired meta-heuristic methods like GA,
D. Rajeswari, V. J. Senthilkumar
ACO and SA for scheduling tasks in a grid computing environment by using single and multi-objective optimi-
zati on wa s done b y Abr aham et al. [16]. Xhafa et al. [17] implemented GA based sched uler. All the above me-
thods convert the multi-objective optimization problem into a scalar cost function, which makes single objective
before optimization.
To minimize the amount of ti me to find the best op timal schedule Li m et al. [18] implemented PGA, Durillo
et al. [19] implemented p arallel execution of NSG A-II and these methods have the d ifficulties to make co mmu-
nication and synchronization between the resources in a distributed environment. This paper implements
NSGA-II with MapReduce programming model and DPGA with MapReduce programming model to find the
best optimal schedule. It makes the task scheduling as an efficient real time multi-objective optimization prob-
3. Multi-Objective Optimization
The Scheduling problem in a distributed environment needs to optimize the several objectives at the same time.
In general, these objectives are contradictory with each other. These contradictory objective functions generates
a set of optimal sol utions. I n the opti mal solution se t, not one so lution is greate r than eac h another sol ution with
regarding all the o bjective functions. T hese opti mal solution set is called as Pareto optimal solution. T he multi-
objective minimization problem is formulated as,
( )( )( )
( )
Minimize ,,, subject to
zgagagaa s= ∈
[ ]
a aaa=
is the vector of decision variables,
gℜ →ℜ
is the objective
functions a nd
is the suitable region in the decision space. A result
is said to do minate another
resul t
, if the subse quent things are satisfied,
{}()( )
{}()( )
kmga gb
kmga gb
∀∈ ≤
∃∈ <
a is said to be a Pareto optimal solution, where no solution dominates
. The b e st solution of the Pareto op-
timal set is called as a Pareto optimal front in multi objective problem space. Once the entire Pareto optimal so-
lution is found, that is the indication of completin g multi-objective problem [20].
Multi objective optimization is also known as vector optimization, as a vector of objectives is optimized in-
stead of a single objective. When using multiple objectives, the search space is divided into two non overlapping
regions kno wn as opti mal and non-optimal. The difference between single objective and multiple objective op-
timization are handling two search spaces and having two goals instead of single.
4. Problem Statement
Distributed environment has geographically distributed computing systems with complex combinations of
hardware, software and network components. R is the set of m processing elements in distributed environment
and T is the set of n tasks assigned to the processing elements. As scheduling is performed for independent tasks,
there is no communication among the tasks and a task can be assigned to a processing element exclusively. The
pre-emption of task is not allowed. As scheduling is performed statically, computing capacity and prior load of
processing ele ment and co mp utational load o f the tas k is estimated a nd the tasks are sc heduled in batc hes, once
allocated the tasks it cannot be migrated to another resource. Expected time to compute matrix (ETC) can be
built by usin g these details. An ET C matrix is a p × q matrix, in which eac h position of t he matrix, ET C [p] [q]
illustrates the expected time to co mplete job p in resource q. The row of ETC matrix has the completion time of
a job of each resource and each column specifies the estimated execution time of a resource for all the jobs.
Hence, the proposed method is static, non-pre e mpti ve sc heduli ng.
In this paper, the objective considered is minimization of makespan and flow time. Make span is the comple-
tion time and waiting time of a task in a processing element. Flow time is defined as the sum of completion time
of all the tasks de scribed as follows,
{ }
min max
sScht taskst
makespan F
D. Rajeswari, V. J. Senthilkumar
{ }
min j
flowtime F
= ∑
whe re
the completion time of task t, tasks stands for a set of all tasks, Sch is se t of all possible schedules.
Longest job has to be scheduled on fastest resource to minimize the makespan and for minimizing flow time,
shortest job to be scheduled on fastest resources. This contradiction makes the problem as multi-objective.
5. Multi-Objective GA with MapReduce for Scheduling Tasks to the
Distributed Environment
5.1. Elitist Non-Dominated Sorting Genetic Algorithm (NSGA-II)
Initial populations of size N are generated randomly. Non-dominated sorting is performed on the population to
classify it into a number of fronts. Crowded tournament selection is performed by assigning crowding distance.
This is used to select a better ranked solution if they belong to differe nt front or select the higher crowding distance
sol ution i f the y belo ng to same front. Crosso ver and mutat ion are per formed o n the generated parent solutio n and
produce the offspring with size N. Single point crossover and swap mutation is used. The parent and offspring
population of size N are combined and produce 2N population. Upd ate the populati on by the indivi duals fr om low-
er front to size n. The indi vidual has small cro wding dis tance will be dropped in t he tie. Precede again, except the
first step till meets the stopping criteria [21]. Figure 1 shows the workflow of NSGA-II with MapReduce model.
Non-dominated sorting: It is used to find the individuals to the next iteration by classifying the population.
The proc edure is given below [22].
Step 1: For ind ivid ual solution p in population N.
Step 2: For ind ivid ual solution q in population N.
Step 3: If p and q are not equal,
Compare p and q for all the objectives.
Step 4: For any p, q is dominated by p, mark solution q as dominated.
First non-dominated set is formed from unmarked solutions.
Step 5: Repe at the procedure till the entire pop ulation is divided into fronts.
Selection: The Crowded tournament selection operator is used. An individual i win the tournament with
another i ndividual j, if one of t he following is true [22].
1. An individual i have better rank, i.e ., ra nki < rankj.
2. T he ind ividua l i and j have the sa me ran k (r anki = r ankj ), then t he i ndivid ua l i has b ette r cr owdin g dist anc e
(in less crowded areas, i.e., di = dj) than individual j.
Crow ding dista nce calcula tion: To break the tie between the individuals are having the same rank crowding
distance is used [22]. The procedure is as follows,
Step 1: Initialize the number of indi vid uals (x) in the front (Fa).
Step 2: Set the crowding dista nce di = 0,
1, 2,,ix=
Step 3: Sort the individuals (x) in front (Fa) based on the objective function (obj). obj =
1, 2,,m
. m is the
number of objectives and S = sort (Fa, obj).
Step 4: Set the distance of boundary individuals as S(d1) = ∞ and S(dx) = ∞.
Step 5: Set k = 2 to (x-1) and calculate S(d2) = S(dx-1) as follows
( )
( )
( )( )
max min
kk mm
Sd Sdff
+ −−
= +
Sk f
is the kth indi vidua l value in S for mth objecti ve function.
5.2. Distance-Based Pareto Genetic Algorithm (DPGA)
At first distance-based fitness assignment was proposed by Osyczka and Kundu [6], the idea gi ving impor tance
to b o th P a re to op timal fr o nt a nd the d i ver si t y of t ha t fr o nt i n o ne fitnes s mea sur e. DP GA use s a d ist a nce compu-
tation and dominance test procedure with complexity O (kη2) [4]. DPGA maintains a standard GA population
and Elite population. The genetic operations like selection, crossover and mutation are performed on GA popu-
lation. All the non-dominated solutions are maintained in elite p opulation. T he step s are as follows [23]:
Step 1: Initialize the r ando m pop ulation of size N (P0)
D. Rajeswari, V. J. Senthilkumar
Figure 1. F lo wc hart for NSGA-II with MapReduce.
Step 2: Calculate the fitness for the first solution (F1) and set generation counter c = 0.
Step 3: If c = 0, insert the first element of P0 in the elite set E0 and perform the distance calculation, mini-
mum distance, index of elite member, elite population updating for each member of the population (k≥2) and
k≥1 for c > 0.
3a. Distance Calculation
Distance calculation is to find the distance between population member and elite member in the objective
space [23]. The formula for calculating distance is given belo w:
( )()
( )
em is the fitness of elite member, fm is the fitness o f population member for particular objectives.
3b. Minimum Distance Calculation
Find the minimu m distance o f a population member co mpared to all the member of an elite set [23]. It is cal-
culated as follows,
min 1
k ik
E is an elite set, dk is the calculated distance.
3c. Elite member ind ex
The Elite member index is used to find which member of elite set has near to the member of the population
{ }
* min
k kk
iid d= =
is the calculated distance and
is the minimum distance.
3d. Fitness Calculation and Elite set up date
The elite set is updated depends on the domination of population member [23]. Population member (k) is
Map phase
Reduce phase
Initialize pop ulation
Fitness function evaluation
Non-dominated sorting
Crowded tournament selection
Crossover and mutation
Fitness evaluation
Select N individuals
Combine parent and child
populat ion, non-dominated
Fitness evaluationFitness evaluation
Aggregate fitness va lue
D. Rajeswari, V. J. Senthilkumar
dominated by elite solution, th e fitness of k is
max 0,k
FFe d
= −
Oth er wi s e
F Fed
= +
and the population member is incl ud e d by eliminatin g all d ominated elite members in elite set.
Step 4: Find the minimum fitness value among all the members of elite population and all elite solutions are
assigned fit ness Fmin.
4a. Minimum fitnes s
Minimum fitness is used to find minimum fitness value among all the members of elite population and all
elite solutions are a ssigned a fitnes s Fmin [23] is,
min 1
min c
E is elite set; F is the fit ness of elite solution.
Step 5: Stop c + 1 = cmax or ter mination criterion i s satisfied. Otherwise, go to step 6.
Step 6: Generate new population (Pc+1) by using selection, crossover and mutation on Pc. Set c = c + 1 and
go to step 3.
There is no genetic operations like reproduction, crossover and mutation is performed on elite population Ec
explicitly. To generate the new population selection, crossover and mutation operators are used. In this paper,
tournament selection, single point crossover and swap mutation are used. Figure 2 shows the workflow of
DPGA with MapReduce model.
5.3. MapReduce Model
Hadoop: Hadoop is an open source framework that provides reliable, scalable, distributed processing and
storage on large clusters of inexpensive servers [24]. Hadoop is written in Java and users can customize the code
to parallelize the data process in clusters which contains thousands of commodity servers. The response time
Figure 2. Flowchart for DPGA with MapReduce.
Map phase
Reduce phase
Initialize pop ulation
Fitness function evaluation
Crossover and mutation
Fitness assignment
Minimum fitness
Fitness assignmentFitness assignment
Aggregate fitness va lue
D. Rajeswari, V. J. Senthilkumar
depends on the complexity of process and volumes of data. The advantage of Hadoop is its massive scalability.
The Apache Hadoop framework consists of Hadoop kernel, MapReduce, Hadoop Distributed File System
(HDFS) and some related projects like HBase, Hive and Zookeeper. At present, Hadoop plays a major role in
Email spa m detection, search engines, ge nome manipulatio n in life scie nces, advertisin g, predic tion in financ ial
service and analysis of log files. Linux and Windows are a preferred operating system for Hadoop.
HDFS: A file system component of Had oop is called a s HDFS. Distributed low-cost hardware is used to store
data in HDFS.HDFS contains name node and data node. A name node has Meta data information. If there is a
request to read a data from HDFS, the name node provides the location of data blocks. The name node also has
overall system information. So the name node is called as master of HDFS. The secondary name node has the
replication of Meta data. At first, data node has to be registered in name node and gets namespace ID. For every
particular period of time, the data node updates its status to name node. HDFS splits the large file into blocks
and stor ed it in the different data nodes. Each block is replicated at the nodes of Hadoop cluster. At the time of
failure data is re-replicated b y the active Hadoop monitoring system [25].
MapReduce programming: It is a distributed parallel processing of large volume of data in a cluster with
fault tolerant and reliable manner. MapReduce has job tracker and task tracker. Job tracker splits the job into
tasks and schedules it to task tracker. The job tracker monitors the progress of tas k tracker. It is also respo nsible
for re-executing the failed tasks. The map phase splits a user program into sub tasks and generates a set of
key-value pairs. It will be submitted to reduce after a shuffle. The reduce phase performs user supplied reduce
function on same key values to generate single entity. The reduce phase is also called as merge phase. The
workflow of MapReduce is presented in Figure 3. The MapReduce function is represented as,
map:: (input _ r e c ord) => list (key, values).
reduce:: (key, list (values) => key, aggregate (values).
5.3.1. NS GA-II with MapReduce
The fitness evaluation of offspring alone has done parallel in the workers available in the map phase.
Non-dominated sorting, crowded tournament selection is performed on an entire population. So it cannot be fit
into concurrent process. 1) Initial population is loaded into coordinator. 2) Coordinator evaluates the fitness
value; perform non-dominated sorting, crowded tournament selection, crossover and mutation. The offspring
generated by coordinator sends to job tracker. 3) The job tracker splits the offspring population and send to
workers of map phase to evaluate fitness value in parallel manner and send it to reduce phase. 4) Shuffle opera-
tion is performed between map and reduce phase. 5) The workers of reduce phase aggregate the fitness value
and send it to the coordinator.
Figure 3. Workflow on MapReduce and HDFS.
Job TrackerName Node
Map-Reduce HDFS
Task Tracker
Data Node
Task Tracker
Data Node
Task Tracker
Data Node
Salve 1Salve 2Salve n
D. Rajeswari, V. J. Senthilkumar
5.3.2. DPGA with MapReduce
This approach can parallelize the distance calculation, minimum distance calculation, elite member index and
fitness calculation. T he evaluation of individuals is executed par allel, because the fitness calculation is indepen-
dent from others in a population. 1) The initial population is seeded into the coordinator 2) The coordinator
produces the offspring population 3) The job tracker divides the offspring population into sub populations and
assigns to workers in the map phase 4) The workers perform the distance calculation, minimum distance calcu-
latio n, el ite me mbe r i nd ex and fit ne s s e va l uat io n for a ss i gne d ind ivi d ua ls c oncurr ent l y 5 ) The worke rs o f r ed uc e
phase collect the fitness values and perform merge operation then send it back to coord inator for the next gene r-
6. Simulation Results and Discussion
The proposed DPGA with MapReduce model is carried out. To estimate the efficie ncy, N SGA-II with MapRe-
duce model is also implemented. The metrics considered for performance evaluation are execution time, ma-
kespan and flowtime.
6.1. Simulation Environment
The simulation is designed by writing programs in Hadoop MapReduce using Java. Hadoop1.2.1 stable version
is used to set up 4 node cluster, which is backed up by HDFS. Hadoop cluster is running on Ubuntu Linux plat-
form and Java 1.6 is used for writing the code. All the 4 systems have i5 processors, 4GB RAM and 500GB hard
disk. The proposed method is evaluated based on factors available in Table 1. Rand o m gener atio n wit h uni for m
distribution i s used for simula tion. Resources can execute a task at a time. ETC matrix is generated depends on
three metrics: task and resource heterogeneity and consistency. The various instances are labeled as x-yy-zz that
represented as follows ,
x-consistency type (c o—consistent, ic—inconsistent, scsemi con sistent).
yy-task heterogeneity (hihigh, lo —low).
zz-processor heterogene i ty (hi—high, lo—low).
6.2. Result Discussions
The algorithm DPGA with MapReduce and NSGA-II with MapReduce model apply to all 12 problem instances.
To compare the performance of multi-objective scheduling algorithm in a distributed computing environment,
the Pareto optimal solutions p roduced by the t wo methods are plotted in Figures 4-7 for all the instances. For a
better comparison of both the algorithm, each was run 10 times repeatedly with various random seeds and the
best solutions are considered for b o th DPGA with MapReduce and NSGA-II with MapReduce.
The makespan and mean flowtime are deliberate in same time units and obtained Pareto optimal solutions are
plotted on a scale of ten thousands of time unit. The plotted graphs indicate that DPGA with MapReduce pro-
duces best schedule in terms of the minimization of makespan and flowtime for all cases compared to NSGA-II
MapReduce method. The algorithms are run for 1000 iterations and 100 initial populations were taken. It is also
noted that the number of solutions obtained in DPGA with MapReduce increases by increasing number of
Table 1 . Sp ecification setting.
Specifications NSGA-II DPGA
Population size 200 200
Number of iteration 1000 1000
Crossover probability (pc) 0.8 0.8
Mutation pro babili ty (pm) 0.01 0.01
Crossover type Si ngle point Single point
Mutation type Swap S wa p
Selection type Cr owded tournament Large tournament
D. Rajeswari, V. J. Senthilkumar
(a) (b)
Figure 4. NSGA-II with MapReduce and DPGA with MapReduce comparison for low task, low processor heterogeneity. (a)
Consistent; (b) Semi consistent; (c) Inconsistent.
(b) (c)
Figure 5. NSGA-II with MapReduce and DPGA with MapReduce comparison for low task, high processor heterogeneity. (a)
Consistent; (b) Semi consistent; (c) Inconsistent.
0.5 0.60.7
mean flowtime
c_lo_lo(N SGA-II)
0.5 0.6 0.7
mean flowtime
s_lo_lo(NS GA-II)
s_lo_lo(DPGA )
0.5 0.6 0.7
mean flowtime
i_lo_lo(N SGA-II)
30 35 40 45
mean flowtime
c_lo_hi(N SGA-II)
36 41 46
mean flowtime
s_lo_hi(N SGA -II_globa l)
s_lo_hi(DP GA)
45 50 55
mean flowtime
i_lo_hi (N SGA-II_glob al)
D. Rajeswari, V. J. Senthilkumar
(b) (c)
Figure 6. NSGA-II with MapReduce and DPGA with MapReduce comparison for high task, low processor heterogeneity. (a)
Consistent; (b) Semi consistent; (c) Inconsistent.
(b) (c)
Figure 7. NSGA-II with MapReduce and DPGA with MapReduce comparison for high task, high processor heterogeneity.
(a) Consistent; (b) Semi consistent; (c) Inconsistent.
18 20 22 24
mean flowtime
c_hi_lo(N S GA-II)
17 18 19
mean flowtime
s_hi_lo(N SGA-II)
s_hi_lo(DP GA)
17 18 19
mean flowtime
i_hi_lo (NSGA-II)
1000 1150 1300
mean flowtime
c_hi_hi( NSGA-II)
1100 1200 1300 1400
mean flowtime
s_hi_hi(N S GA-II)
s_hi_hi(D PGA )
1100 1300 1500
mean flowtime
i_hi_hi( NSGA-II)
D. Rajeswari, V. J. Senthilkumar
population and number of iterations. Figures 4-7 show that DPGA with MapReduce model generates better
makespan and mean flowtime for all consistency type, task and processor heterogeneity.
6.3. Performance Comparison of NSGA-II and DPGA
The P areto optimal sol ution s et o btained b y NSG A-II, DPGA satisfy different objectives to so me extend [26]. A
fuzzy based technique is used to select best compromise solution from the attained non-dominated set of solu-
tions [27]. The fuzzy sets are defined using triangular membership function. Consider fma x and fmi n are maximum
and minimum values of each objective function, solution in the kth objective function of a Pareto set fk is de-
scribed by a membership function µk illustrated as,
max min max
max min
kk kk
fff ff
= <<
The value of membership function indicates ho w far a non-dominated solution has satisfied the objective. In
order to measure the performance of each solution to satisfy the objective, the sum of membership function val-
ues µk is computed, where
objectives. The performance of each non-dominated solution can be
rated with respect to the entire N non-dominated solutions by normalizing its performance over the sum of the
ability of N non-dominated solutions as follows,
inm i
= =
where n is the a mount of sol ut ion a nd m is the amo unt of ob j ect ive fu nctio ns. The solu tio n has t he b est val ue of
µi is the soluti on. T he ma kesp an and mean flow t ime val ue fo r the fi net ad jus tment solut ion at taine d is l isted i n
Table 2. The percentage of reduction in makespan and mean flow time of DPGA with MapReduce over
NSGA-II with MapReduce is calculated as,
percentage of makespan reduction1100
mean flowtime
percentage of mean flowtime reduction1100
mean flowtime
The makespan and mean flow time reduction percentage is present in the Table 2. DPGA with MapReduce
achieves a reduction in makespan and flowtime by 12% and 14%, over the values of NSGA-II with MapReduce.
Figures 4-7 also indicates DPGA with MapReduce outperfor ms over NSGA-II with MapReduce.
6.4. Execution Time Comparison of NSGA-II and DPGA
NSGA-II with MapReduce and DPGA with MapReduce is executed in single node and 4 node Hadoop cluster.
The time taken by both the algorithms to find the optimal schedule is listed in Table 3. It is noted that DPGA
with MapReduce has less execution time than NSGA-II with MapReduce. As the number of nodes in a Hadoop
cluster is increased, the execution time of these algorithms will be reduced.
7. Conclusion
In distributed computing s ystems, allo catio n of ta sks to the p ro cessing ele ment with minimu m amount o f time i s
a key step for better utilization of resources. In this paper, NSGA-II with MapReduce and DPGA with MapRe-
duce is implemented in DS environment and their makespan, mean flow time and execution time are compared.
From the obtained results, it is noted that DPGA with MapReduce model achieves a reduction in makespan,
mean flow time and execution time by 12%, 14% and 13%. The simulation results also show that the execution
D. Rajeswari, V. J. Senthilkumar
Table 2 . Comparison of NSGA-II with MapReduce and DPGA wit h MapRedu ce.
Instances NSGA_II with Ma pRe duce and Fuzzy DPGA with MapRe duce and Fuzzy Percentage o f
Reduction in
makes pan by DPGA
Percentage of
Reduction in mean
flow time b y DPGA
Makespan Mean flowtime Makespan Mean flowtime
co_lo_lo 6546.28 90838. 43 5748.18 85539.56 12.19 5.83
co_lo_hi 376851.95 5133760.26 3 13263.46 4354684.62 16.87 15.18
co_hi_lo 216064.67 2846845.19 194562.67 2605329.13 9.95 8.48
co_hi_hi 12548739.42 152960763.32 10965632.48 142659273.5 12.62 6.73
sc_lo_lo 6145.27 80376.11 5259.03 70540.74 1 4.42 12.24
sc_ lo_hi 449285.18 5304352.69 388432.8 4 146759.85 13.54 21.82
sc_hi_lo 184273 .52 2368543.50 170501.59 2175234.38 7.47 8.16
sc_h i_hi 12864789.63 160436981.84 11259642.43 106847994.9 12.48 33.40
ic_lo_lo 6389.37 76943.33 5346.35 68259.43 16.32 11.29
ic_lo_hi 522843.21 5679384.21 467899.59 4938345.23 10.51 13.05
ic_hi_lo 180437.35 2344622.15 171243 .62 2132541.59 5.10 9.05
ic_hi_hi 13901746.88 167435439.5 1191359 2.13 138565274.3 14.30 17.24
Table 3. Comparison of execution time.
Methods NSGA-I I with MapReduce DPGA with MapR educe Percentage of redu ction
Sequential (Single node) 34.68 Sec 31.93 Sec 7.93
Parallelism w ith MapReduce
(4 node Cluster) 31.714 Sec 27.64 Sec 12.85
time of these a lgorithms will be reduce d, while increasing the number of nodes in a Hadoop cluster. Future work
could be extended for implementing a scheduler by using all the evolutionary kind of algorithms with the Ma-
pReduce model to execute its p arallel without coordination issue.
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