Fuzzy Based Evaluation of Optimal Service Time Using Execution/Communication Time and Clustering
INTRODUCTION
Various applications of assignment problem (AP) exist in the real world. Extensive examples can be seen in various areas of the assignment problem such as industry, commerce, technology, management science, etc. Reallife problems cannot be successfully solved by the classical optimal assignment problem. Fuzzy assignment problem (FAP) is more appropriate in today’s context. The theory of fuzzy sets was given in 1965 by Lofty A. Jadeh. This theory represents impurity or uncertainty in daily life. With the increase in computer processing, the need for competencies in technology has increased even more rapidly. Expected handling effectiveness for special applications cannot be attained with a monotonous processing framework. These problems can be solved by distributed processing systems (DPS). The division and allocation of work are important in building DPS. If these steps are not executed properly, the structure of DPS increases the number of promoters that can reduce the overall flow of the structure. With advances in distributed systems, the assignment problem has turned into an important consideration. There is a need to improve efficiency for computing scheduling in broadcasting work. The most important topic when structuring any task algorithm is to minimize the extra time as much as possible.
An optimal solution is obtained by calculating a fuzzy mean service rate using the task allocation technique (Khandelwal, 2019) with the idea of fuzzy communication time. The author (Yadav et al„ 2019) has allocated the processors efficiently on different nodes by balancing the load between them. In this process, care has also been taken to reduce execution and response time. An optimal solution to the TFA problem has been achieved using the centroid ranking technique (Mary and Selvi, 2018). In addition, it uses the Euclidean separation strategy to analyze the fuzzy value and its rank. Many researchers have solved the fuzzy assignment problem through a genetic algorithm approach. Through the genetic algorithm approach, each individual with cost (time) has been organized for only one task (Muruganandam and Hema, 2018) and is presented as an invariant number. In addition to the genetic algorithm approach here, crisp values have been derived by the Yager ranking method to obtain an optimal solution to the fuzzy assignment problem.
In various research papers, the defuzzification concept (Sharma, 2018. Kumar et al„ 2013, Gani and Mohamed, 2013 and Yadav et ah, 2011) is envisaged by the robust ranking method, wherein the fuzzy value is changed to one, i.e., crisp number one. In defuzzification, a mathematical model is formed to determine the correct response time of the system using triangular/trapezoidal fuzzy execution time and triangu lar/trapezoidal fuzzy intertask communication time. Fuzzy assignment problem is important in solving the reallife problem. In the reallife problem (Thakre et ah, 2018, Selvi et ah, 2017), the fuzzy assignment problem has been resolved through the example of four candidates/designations at Life Insurance Corporation (LIC). In this the solution of this problem is represented by the fuzzy triangular magnitude ranking method, Hungarian method, MOA and direct method. A diffusion strategy (Neelakantan and Sreekanth, 2016) has been proposed to accommodate tasks between different computers in the context of a problem. In this, the response time of assignment to a processor is limited from moving the overload computer to the lowload computer to using the load adjustment system from the computer. To assess the performance of the system, the load on the system is calculated by fluctuating the average differential arrival time of the tasks on each computer.
PROBLEM STATEMENT
Fuzzy Assignment Problem (FAP) is more appropriate for today’s reallife problems. Here, we have a fuzzy assignment problem (FAP). The purpose of this fuzzy assignment problem is to allocate a set of processors F_{:} [pj} on a set of tasks F_{:} {f,} in such a way that the total impedance time is minimized. For the solution of the fuzzy assignment problem here we convert the fuzzy impedance time coefficients by using the fuzzy ranking method and then formed clusters of tasks by using Kmeans clustering technique.
Here, execution time, communication time, total response (impedance) time, defuzzification, and fuzzy mean service rate (Khandelwal, 2019) have been solved by using the following formulae.
• Fuzzy Execution Time:
• Fuzzy Communication Time:
• Fuzzy Total Response Time:
• Defuzzification:
• Clustering Technique: Clustering technique aims to partition m tasks into n cluster (m > n) in which each task belongs to the cluster with the adjacent mean. This technique produces exactly n different clusters of greatest possible discrepancy.
• Fuzzy Mean Service Rate:
COMPUTATIONAL ALGORITHM
The mapping between the tasks and processors is defined by (p: N —» M. A task may be a data file or code which is to be executed on different processors having different processing capabilities. Assume that number of tasks is more than the number of processors (m > n) as normally seen in real life. Also it is assumed that the execution time of a task on each processor and intertask communication time is known. The intertask communication time between the same tasks is zero.
Step 1: Set Fuzzy quantitative problem of m tasks F {f, j for l
Step 2: Set Fuzzy quantitative problem of n processors F_{:} {p,} for 1< j < n, le.,{p,,p_{2},p_{3}}
Step 3: Set F_{:} {/f7(cT_{v} )}and F, {CT(ct_{ik})} in the form of fuzzy triangular number. F {£T(et,y)}and F {C7(c?_{rt})}are taken in the form of matrices as Fuzzy Execution Time Matrix and Fuzzy Intertask Communication Time Matrix.
Step 4: Determine Defuzzified Crisp Value for F_{;} {ETfc^)}by using Eq. (11.4).
Step 5: Determine sum of each task processorwise and stored in Fuzzy Sum Array F_{Z}{ET_{S},_{U}„_«„*{}}.
Step 6: Cluster the task into n processor.
Step 7: Select n points randomly as cluster centre and calculate the squared error function using Eq. (11.5).
Step 8: Assign task to their closest cluster centre.
Step 9: Calculate mean of all tasks in each cluster.
Step 10: Repeat steps 7. 8 and 9 until the same points are assigned to each cluster in consecutive rounds.
Step 11: Apply basic assignment method on these clusters so formed in step 10 and allocate these tasks clusters on different processors.
Step 12: Let tasks allocated to processors is denoted by F CTET(ety)} = T_{a},,„_{cah}.. Calculate Total Fuzzy Execution Time F {ET}, Total Fuzzy Communication Time F{CT} and Total Fuzzy Task Response Time F{TRT}using Eqs.
(11.1), (11.2), and (11.3).
Step 13: Convert Total Task Response Time into crisp values once using step4 because it is fuzzy quantitative number.
Step 14: Now' calculate the Overall Task Response Time for all tasks allocated in all different processors and Fuzzy Mean Service Rate for each Processor F_{:}{MSR(,)}
Step 15: Stop.
PSEUDOCODE FOR METHOD
Begin
 • Set F_{z} {tijwhere 1 < i < m and F_{z} denotes fuzzy allocation for task {ti}
 • Set F_{z} {pj} where 1 < j < n
 • Set Fuzzy Triangular Number F_{z} {ET(et_{i;(})} and F_{z} {CT(ct_{lk})}
 • Represent F_{z} {ET(et_{i;l})} and F_{z} {CT(ct_{iic})} in rectangular matrix form (m > n)
 • for i: =1 to m inclusive do
for j:=1 to n inclusive do
1 .
Calculate, R(Cj,) = — (a + 3a
2 Jo
F_{z} {ET(et_{1;})} < Mag(,Cij) end for end for
• Step5: sum = 0, a =
for i:=1 to m inclusive do for j:=1 to n inclusive do
if (((i % 2==0) (i % 2==l)) && (j<=n)) then sum = et[i] [j]+sum end if end for sum_{a} = sum
sum = 0
a = a + 1
end for
• for i: =1 to m inclusive do
for j:=1 to n inclusive do T(i, j) = (tj % Cj) end for end for
for i:=1 to m inclusive do T(i)T(i)array)_{min}
Cl] = compare [diff(Tk_{k+lk+rm}._{ni} where k + r=j, 1 < j < n_{;}
r = 0,1, 2,...
then again
( (C* ^ T(i)k + lmin_{m}2_{n}.......I (Cfc+r ^ IXiJmin
end for
 • Calculate mean of all tasks {ti}in each cluster {Clj}
 • Repeat steps 5, 6 and 7 until the same points are assigned to same cluster {Clj} in consecutive rounds.
 • Apply Hungarian method on these clusters so formed in step 8 and allocate
{Pj} < (tj(Clj) }.
 • Set F_{z} ) = ^2locate * {ti(Clj) }
 • Calculate Total Fuzzy Execution Time, f_{z} {et(c)} =
Total Fuzzy Communication Time, ^{lsism}
F_{z} {cr (c)} = {ct_{iit}} , к = 1, 2, 3, . . . , Ш ; 1 < j < n
i=j*k
• Calculate overall FTRT for all tasks,
F_{z} {trt(c)} = max (F_{z} {et(c)} + F_{z} {CT(c)})
 1
 • Convert, f_{z} {trt(c)} < i J(a + 3a_{0}  a)f(k)dk
 0
^TtA
 • Calculate, f_{z} {MSR(j)} = ———
 1 ^{J} ' F_{z}{ET(etj) }
End
MATHEMATICAL IMPLEMENTATION
Steps 13: Set Fuzzy numerical problem for m tasks for 1
i.e., {fj,T_{2}}and n processors F_{;}{/ty} for 1< j
TABLE 11.1
Fuzzy Execution Time Matrix
(5,10.20) 
(10,15.20) 
(10,20,30) 
(5.10,20) 
(5,10,15) 

(5,10.15) 
(10,20,30) 
(10,15.25) 
(10,15.20) 
(5,10,20) 

(10,15,20) 
(10,15,25) 
(10.15,20) 
(5,10,15) 
(5,15.20) 
TABLE 11.2 Fuzzy Communication Time Matrix
(0,0,0) 
17.5 
10 
31.25 
6.25 

17.5 
(0,0,0) 
32.5 
10 
25 

10 
32.5 
(0,0,0) 
10 
10 

31.25 
10 
10 
(0,0,0) 
12.5 

6.25 
25 
10 
12.5 
(0,0,0) 
Step 4: Using Ranking Method find Crisp Value for F {/:T((?/,_{;})}obtained in Table 11.3.
TABLE 11.3
Defuzzified Crisp Values
6.25 
8.75 
10 
6.25 
5 

5 
10 
10 
8.75 
6.25 

8.75 
10 
8.75 
5 
5 
Step 5: Determine sum of each task (processorwise) and stored in Fuzzy Sum Array F {ET_{Sum} Row shown in Table 11.4.
TABLE 11.4
Fuzzy Task_Sum Array
Task 

Crisp Value 
20 
28.75 
28.75 
20 
16.25 
Step 6: Cluster the task into three processors.
Steps 710: Select three points {18,22,26} randomly as cluster centre and calculate the squared error function (by Eq. [11.5]). Tables 11.511.7 representing the stepwise clustering iterations.
TABLE 11.5
Centred Mean Value in Three Different Iterations
18/18.125/16.25 
22 / 20,.з 
26 / 28.75_{2}.з 

18/18.125/16.25 
22 / 20,.з 
26 / 28.75_{2}.з 

18/18.125/16.25 
22 / 202,з 
26 / 28.75_{2}.з 

18/18.125/16.25 
22 / 202,з 
26/28.75_{2}.з 

18/18.125/16.25 
22 / 202.з 
26/28.75_{2}.з 
TABLE 11.6
Deviation During Clustering in Three Different Iterations
2/1.875/3.75 
2 / 0_{2},з 
6/8.75_{2}.з 

10.75/10.625/12.25 
6.75/8.752.3 
2.75 / 0_{2}.з 

10.75/10.625/12.25 
6.75/8.752.3 
2.75 / 0_{2}.з 

2/1.875/3.75 
2 / 0_{2},з 
6/8.752.3 

1.75/1.875/0 
5.75/3.75_{2}.з 
9.75 / 12.5_{2}.з 
TABLE 11.7
Nearest Cluster with New Centroid Value in Different Iterations
Nearest Cluster 
NewCentroid 

x/2/2 
20/20*/20* 

3/3/3 
28.75“ / 28.75“ / 28.75“ 

3/3/3 

3/3/3 
18.125 /* /*/ 

1/1/1 
18.125“ /* /16.25 
In 2nd and 3rd iteration same points are assigned to each cluster. Hence tasks clustered on three processors as /_{:} ®/_{4}},{/_{2} Ф/,}{/_{5}} in three consecutive rounds.
Step 11: Apply basic Assignment procedure to allocate clustered tasks on the different processors. It is shown in Table 11.8.
TABLE 11.8
Assignment Method on Cluster
12.5 
18.75 
5 

13.75 
20 
6.25 

13.75 
18.75 
5 
Steps 1214: Evaluate F {£T}, F_{:} {C7^{1}} and F_{z} {ТЯГ} using Eqs. (11.1), (11.2), and (11.3). Also Evaluate Fuzzy F {MS/?(,)}fcr each Processor. These values are show n in Table 11.9.
Step 15: Stop
TABLE 11.9
Fuzzy Total Response Time and Mean Service Rate
Processor 
Tasks 

12.5 
(70,120.175) 
(80,140,215) 
78.5 
0.0254 

6.25 
(60,95,130) 
(65,105,145) 
58.75 
0.0170 

18.75 
(90,145,205) 
(110,180.255) 
101.25 
0.0197 
CONCLUSION AND DISCUSSION
In this research problem, we have established a fuzzy system using fuzzy execution time and fuzzy intertask communication time through which the optimal value of the mechanism can be obtained. Here, tasks are moved from overload processors to underload processors using a mixture of functions to reduce the average service rate of tasks. To estimate the optimal value of the system according to the process, different tasks have been allocated according to the average service rate of the tasks on each processor. Compared to existing approaches, the proposed technique has yielded the optimal value minimum. The proposed approach reduces the average service rate of the system for an effective mix of tasks and optimal allocation, which is an important part of this technique. According to the approach proposed here the optimum value is 101.25 w'hich is lower than the 66.83 and 65.16 values as compared to the existing approaches.
The methodology proposed in this chapter qualifies for the allocation of the processor’s functions and provides optimal values, but we cannot say that this currently existing technique provides perfection to this mechanism. We can use various other methods of solving various technology in this field. Here, we have only focused on reducing the average service rate of the model, but this may also apply to load balancing problems w'ith different models of phased technology in the future.
REFERENCES
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