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A NEW RANKING IN HEPTAGONAL FUZZY NUMBER AND ITS APPLICATION IN PROJECT
SCHEDULING
Adilakshmi Siripurapu1, Ravi Shankar Nowpada2
Dept. of Basic Science and Humanities, Vignan's Institute of Information Technology(A), Duvvada, Visakhapatnam, AP, India1 [email protected] Dept. of Mathematics, Institute of Science, GITAM (Deemed to be University), Visakhapatnam, AP, India2 [email protected]
Abstract
Ranking fuzzy numbers is significant in optimization approaches such as assignment challenges, transportation problems, project schedules, artificial intelligence, data analysis, network flow analysis, an uncertain environment in organizational economics etc. This paper introduces a new fuzzy ranking in Heptagonal fuzzy numbers and arithmetic operations of Heptagonal fuzzy numbers defined. In the network, every activity duration is viewed by a Heptagonal fuzzy number. Every Heptagonal fuzzy number is transformed into a crisp number using the ranking function. By applying the traditional method, we calculate the fuzzy critical path. These procedures are illustrated with numerical examples and compared with existing ranking functions.
Keywords: Activity duration, centroid, fuzzy ranking, fuzzy critical path, heptagonal fuzzy number.
I. Introduction
One of the most significant concepts in network analysis is the critical path approach. It is utilized to resolve project problems by preparing the networks and determining the earliest date an activity may begin and be completed. It is also an algorithm for scheduling a collection of project networks. It is also frequently used in connection with the Program Evaluation and Review Technique (PERT).
Zadeh [10] introduced the 'fuzzy logic' concept considering inaccuracies and inconsistencies. Several academics have utilized various forms of fuzzy numbers to develop mathematical models over the last few decades. Examples for fuzzy numbers include Triangular fuzzy numbers, Trapezoidal fuzzy numbers, Pentagonal fuzzy numbers, etc.
In many practical situations, the variables that define information uncertainty or vagueness are usually Triangular or Trapezoidal fuzzy numbers. Chandrasekaran et al. [1] developed a new arithmetic operation in Heptagonal fuzzy numbers and solved the transportation problem in 2013. Rathi et al. [8] defined a new non-normal fuzzy number called the Heptagonal fuzzy number and
Adilakshmi Siripurapu, Ravi Shankar Nowpada RT&A, No 2 (68) HEPTAGONAL FUZZY NUMBER AND ITS APPLICATION__Volume 17, June 2022
arithmetic computations, suggested a parametric ranking strategy for ordering Heptagonal fuzzy numbers and employed the fuzzy assignment problem in 2014. The Heptagonal fuzzy number ranking is derived from the centroid of centroids and incentre of Heptagonal fuzzy numbers by Namarta et al. in 2017[7]. Developed a new ranking in Heptagonal fuzzy numbers and adapted it to transportation problems by Sahaya et al. [9]. Karthik et al. [4] in 2019 proposed linear and nonlinear Heptagonal fuzzy numbers under uncertain environments and derived the Haar ranking technique for the Heptagonal fuzzy number. Malini [6] suggested a new ranking in Heptagonal fuzzy numbers and applied transportation problems. Hamildon et al. [3] determine the fuzzy critical path with normalized Heptagonal fuzzy data.
II. Preliminaries
In this section, we will look at a few key definitions. I. Fuzzy Set [10]
As stated in Zadeh's paper, the formalization of a fuzzy set is:
Let X be a space of points (objects), with a generic element of X denoted by x. Thus, X = {x}. A fuzzy set (class) A in X is characterized by a membership (characteristic function) function ^A(x), which associates with each point in X a real number in the interval [0,1], with the value of ^A(x) at x representing the "grade of membership" of x in A. When A set in the ordinary sense of the term, its membership function can take on only two values, 0 and 1, (x) = 1 or 0 according to x does or does not belong to A.
II. Fuzzy Number [5]
It is a Fuzzy set of the following conditions:
• Convex fuzzy set
• Normalized fuzzy set.
• Its membership function is piece-wise continuous.
• It is defined in the real number.
Fuzzy numbers should be normalized and convex. Here the condition of normalization implies that the maximum membership value is 1.
III. Heptagonal fuzzy number (HFN) [3]
A fuzzy number A = (a1, a2, a3, a4, a5, a6, a7) is a normal Heptagonal fuzzy number, and its membership function is expressed as;
1 (x-a1)
----, for a, <x<a2
2(02-01)' ' 1 2
0.5, for a2<x<a3
1 . 1 (x-a3)
—, for a3 < x < a4
ßÄ(x) = <
2 + 2 (a4 - a3)' ll(os-x)
2+2(Ö^-J' f0r ^4 0.5, for a5 < x < a6
1 (07-X)
-7-T' for a6 < x < a7
2(07-a6y ' 6 7
O, otherwise
The graphical depiction of normalized Heptagonal fuzzy number is represented in Figure 1.
Figure 1: Graphical representation of Heptagonal fuzzy number
IV. Generalized Heptagonal fuzzy number (GHFN) [3]
A generalized Heptagonal fuzzy number is denoted by Äß = (a1,a2,a3,a4,a5,a6,a7,M) and its membership function is expressed as;
m (x — a1)
2 (a.2 — ai)
, for a1 < x < a2
— , for a.2<x <a3 m m (x — a3)
- + -7-r, for a3<x<a4
2 2 (a4 — a3)
ii~ (x)= u u (a5—x)
var(x)= - + ---—, for a4<x<as
2 2 (as — a4)
— , for a5<x<a6
m (a7 — x)
, for a6 < x < a7
2 (a7 - a6) ^ Otherwise, 0
The graphical depiction of generalized Heptagonal fuzzy number is represented in Figure 2.
Figure 2: Graphical representation of Generalized Heptagonal fuzzy number
V. Arithmetic Operations of Heptagonal fuzzy number
According to Dubois [2], defined the arithmetic operation of Heptagonal fuzzy number.
Let Ä = (a1, a2, a3, a4, a5, a6, a7) and B = (b1, b2, b3, b4, b5, b6, b7) be two Heptagonal fuzzy numbers
then;
kA = k(a1,a2,a3,a4,a5,a6,a7) = (ka1,ka2,ka3,ka4,ka5,ka6,ka7)
A © 8 = (a1 + b1, a2 + b2, a3 + b3, a4 + b4, a5 + b5, a6 + b6, a7 + b7)
AQ B = (a1- b^ a2 — b2, a3 — b3, a4 — b4, a5 — b5, a6 — b6, a7 — b7)
A® 8 = (a1* b^ a2 * b2, a3 * b3, a4 * b4, a5 * b5, a6 * b6, a7 * b7) /a-, a2 a3 a4 a5 a6 a7\
A® B = (-1,-2,-3,-4,-5,-6,-7)
\a.2 b2 b3 b4 b5 b6 b7J
Example:
Let A = (3,6,9,12,15,18,21) and 8 = (2,4,6,8,10,12,14) then
A®B = (5,10,15,20,25,30,35)
AQB = (1,2,3,4,5,6:7)
A®B = (6,24,54,96,150,216,294)
A®B = (1.5,1.5,1.5,1.5,1.5,1.5,1.5)
Remark: Some authors defined A Q 8 = (a1 — b7, a2 — b6, a3 — b5, a4 — b4, a5 — b3, a6 — b2, a7 — ¿1).
How is it possible?
Here I consider one example.
Let A = (2,4,6,8,10,12,14) & 8 = (2,4,6,8,10,12,14) , Here both A and 8 are same HFNs. Now AQ 8 = (2 — 14,4 — 12,6 — 10,8 — 8,10 — 6,12 —4,14 — 2)
= (—12, —8, —4,0,4,8,12). It is a completely wrong output since both A and 8 are the same HFNs. According to my definition;
AQB = (2 —2,4 —4,6 —6,8 —8,10 —10,12 —12,14— 14) = (0,0,0,0,0,0,0)
III. Existing Rankings
In this section, we explained existing ranking functions. I. Existing Ranking1[7]
Namarta et al. suggested a ranking process for HFN prediction on the centroid of centroids and incentre of centroids. Their suggested order is as follows:
(2a1 + 7a2 + 7a3 + 22a4 + 7a5 + 7a6 + 2a7 11w\ G(xo>yo) = (-54-^
II. Existing Ranking 2 [3]
Hamildon et al. identify the critical path of a project network with normal HFNs. In his approach,
a1 + a2+ a3 + 2a4 + a5 + a6 + a7 k(ah) =-g-
IV Proposal Ranking Function
We suggest an effective tool for calculating the rank of HFN. The proposal ranking in a HFN diagram is represented in Figure 3.
Figure 3: Proposal ranking in GHFN
In Figure 3, the heptagonal is divided into two trapezoidal and one rhombus. By applying the centroid formula of trapezoidal and rhombus, calculate the centroid of trapezoidal and rhombus, respectively. The circumcentre of the centroids of the HFN is taken into a balancing point of the Heptagon in Figure 7.3. The distance from the origin to the circumcentre of the centroids of this three-plane figure consider as a generalized HFN ranking function. Let the centroid of the three planar figures be G^ G2, G3.
G1 gives the centroid of the trapezoidal with vertices
G2 gives the centroid of the rhombus with vertices
(a4,0), (a3,f), (a4,м), { a^f) G3 gives the centroid of the trapezoidal with vertices
{0,4,0), (a5^j, {ae,^), (a7,0) The centroid of these three planes is;
fa4 + a5 + a6+a7 .. ^
14 5 6 7,—) respectively.
_ (a1 + a2 + a3 + a4 „ (,
U, - I-:-,4),^ - (
44
The circumcentre of G1, G2, and G3 is
0.3+0.4+0.5+0.4 r _
~4-,J),G3 -
n / \ /1
'Afl{Xo,yo) - (
0,1 + 0,2 + 20,3+40,4 + 20,5 + 0,6+0,7 to 3
■f)-
The generalized Heptagonal fuzzy number Aq — {a1, a2, a3, a4, a5, a6, a7 w) new ranking function is;
«№) -
xl + y02
I. Ordering of a Heptagonal fuzzy number
Comparing fuzzy numbers using the ranking function "ft: F(R) —> K is a successful approach. The ordering of two HFNs is described as follows;
• If K(AS)>K(BS)^AS>BS
• If K(Aff)<K(Bff)^Aff<Bff
• If K(Aff)-K(Bff)^Aff-Bff
Here, we utilize three sets of Heptagonal fuzzy numbers. Analyze the ranking of 3 sets by proposal ranking function and existing ranking functions. The sets and the outcome obtained by the proposal and existing rankings are given in Table 1. Here I consider ra=1.
Table 1: Analysis of ranking order by proposal ranking function
HFN
The rank of HFN
Conclusion
Set-1
A =(1,2,3,4,5,6,7) 4.0138
B =(3,4,5,7,9,10,11) 7.0079
C =(2,3,4,8,13,14,15) 8.3399
D =(1,2,3,8,9,11,13) 6.9246
Set-2
A =(6,8,9,10,11,12,13) 9.9334
B = (3,4,5,6,7,8,9) 6.0277
C = (8,9,10,11,12,13,15) 11.0983
D = (5, 7,8,9,10,11,12) 8.9353
Set-3
A = (5,10,15,22,23,24,25) 19.0087
B = (4,10,12,17,18,19,21 15.1776 ^ > ^ > ^ > m
C = (3,10,12,13,14,16,17) 12.5133 ^A>B>C>D
D = (3,6,8,10,11,12,13) 9.3511
The order of a HFN with the proposal and existing rankings are presented in Table 2.
Table 2: Order of a HFN with the proposal and existing rankings
Ranking function Set-1 Set-2 Set-3
Namarta (2017) C > B > D > A C >A>D>B A > B > C > D
Hamildon (2021) C > B > D > A C >A>D>B A > B > C > D
Proposal ranking C > B > D > A C >A>D>B A > B > C > D
V Application
This section performs an analytical example of the proposed fuzzy set CPM-based approach on an activity network. Think of a plant with 14 vertices and needs 21 primary activities, each activity connected by a direct link, like in the following graph (Figure 1.6). The fuzzy activity time is represented as a Heptagonal fuzzy number for every activity in Table 3 (All durations in days).
Table 3: Project network with Heptagonal fuzzy number Activity Heptagonal Fuzzy Number
1^2 (2,3,4,8,13,14,15)
r(c) > r(b) > k(d) > r(a)
^C>B>D>A
r(c) > r(a) > k(d) > r(b)
^C>A>D>B
1^3 (1,2,4,5,11,12,13)
1^4 (0,1,3,4,9,10,11)
1^5 (1,3,6,13,15,16,17)
1^6 (1,2,5,10,12,14,17)
2^7 (1,2,11,13,14,15,16)
3^7 (2,3,4,8,14,15,16)
3^10 (1,5,7,11,12,13,14)
4^8 (0,1,2,3,5,6,8)
4^9 (1,2,4,5,6,7,9)
5^9 (3,7,9,13,15,16,18)
5^13 (5,10,15,22,23,24,25)
6^14 (1,3,4,5,7,8,10)
7^11 (10,12,15,20,21,22,23)
7^12 (3,5,6,7,8,9,10)
8^12 (7,8,9,10,11,12,13)
9^13 (6,6,6,6,7,7,7)
10^14 (3,4,5,6,7,8,9)
11^14 (4,6,7,9,10,11,12)
12^14 (2,3,4,5,6,7,8)
13^14 (5,7,8,9,10,11,12)
Figure 4: Fuzzy project network
I. Expected time of activities
Heptagonal fuzzy number transformed into an activity duration by proposal ranking function. This activity period is taken as the time within the nodes, and the fuzzy critical path is calculated by applying the conventional process. The activity period of the project network is represented in
Table 4, and Figure 5 represents the project network with defuzzified values of Heptagonal fuzzy numbers.
Activity HFN Activity period (tiy-)
1^2 (2,3,4,8,13,14,15) 8.3399
1^3 (1,2,4,5,11,12,13) 6.5085
1^4 (0,1,3,4,9,10,11) 5.1774
1^5 (1,3,6,13,15,16,17) 10.9217
1^6 (1,2,5,10,12,14,17) 9.0061
2^7 (1,2,11,13,14,15,16) 11.3382
3^7 (2,3,4,8,14,15,16) 8.6730
3^10 (1,5,7,11,12,13,14) 9.5891
4^8 (0,1,2,3,5,6,8) 3.4328
4^9 (1,2,4,5,6,7,9) 4.9279
5^9 (3,7,9,13,15,16,18) 12.0046
5^13 (5,10,15,22,23,24,25) 19.0029
6^14 (1,3,4,5,7,8,10) 5.3437
7^11 (10,12,15,20,21,22,23) 18.2530
7^12 (3,5,6,7,8,9,10) 6.9246
8^12 (7,8,9,10,11,12,13) 10.0055
9^13 (6,6,6,6,7,7,7) 6.3420
10^14 (3,4,5,6,7,8,9) 6.0092
11^14 (4,6,7,9,10,11,12) 8.5898
12^14 (2,3,4,5,6,7,8) 5.0110
13^14 (5,7,8,9,10,11,12) 8.9228
Figure 5: Project network with defuzzified values of HFN
Adilakshmi Siripurapu, Ravi Shankar Nowpada KT&A^ No 2 (68) HEPTAGONAL FUZZY NUMBER AND ITS APPLICATION__Volume 17, June 2022
II. Procedure for Fuzzy critical path method Based on Heptagonal fuzzy numbers Step 1: Construct the network diagram of a given project.
Step 2: Represent every fuzzy activity time as a defuzzified value of the Heptagonal fuzzy number.
Step 3: Let ¡:1 - {0,0,0,0,0,0) and calculate E^, i - 2,3,......,n by using;
Ei - maximum end time of immediate predecessor + activity duration. Step 4: Calculate the earliest finish time of an activity i ^ j. That is earliest finish time is; EFij -E§ij + Fuzzy activity time, where E§ij - E^.
Step 5: Let En - Ln and calculate Lt, where i - n — 1,n — 2,......,2,1.
Li - minimum end time of immediate successor — activity duration
Step 6: Calculate the latest start time of an activity i ^ j. That is latest start time is;
L§ij - LFij — Fuzzy activity time=, where LFij - Lj Step 7: Calculate total float TP^ - LFij — EFij or LS^ — E5>ij.
Step 8: If TFij - 0, consider those activities as critical activities of a given project network.
III. Calculation of Earliest times
Node 1 is the starting node in the above network, and node 14 is the end node. Let E1 - 0, and label node one as 0. Iteration 1:
Node1 is the predecessor of node2.
••• E2- E1 + t12 - 0 + 8.3399 - 8.3399
Iteration 2:
Node1 is the predecessor of node3.
••• E3- Ii1 + i13 - 0 + 6.5085 - 6.5085
Iteration 3:
Node1 is the predecessor of node4.
• E4- ¡i1 + t14-0 + 5.1774 - 5.1774
Iteration 4:
Node1 is the predecessor of node5.
• £5-^ + ^-0 + 10.9217 - 10.9217
Iteration 5:
Node1 is the predecessor of node6.
• £¿-^ + 1^- 0 + 9.0061=9.0061
Iteration 6:
Node2 and node3 are predecessor of node 7.
• E7 - max[E2 + t27,E3 + t37} - E2 + t27 - 8.3 3 99 + 11.3 3 82 - 19.6781
Iteration 7:
Node4 is the predecessor of node8.
• E8-E4 + t48 - 5.1774 + 3.4328 - 8.6102 Iteration 8:
Node4 and node5 are the predecessors of node9.
• E9 - max[E4 + t49,E5 + t59] - E5 + t59 - 10.92 1 7 + 12.00 46 - 22.63 43
Iteration 9:
Node3 is the predecessor of node10.
• E10 - E3 + t310 - 6.5085 + 9.5891 - 16.0976
Iteration 10:
Node7 is the predecessor of node11.
• E11 - E7 + t711 - 19.6781 + 18.2530 - 37.9311
Iteration 11:
Adilakshmi Siripurapu, Ravi Shankar Nowpada RT&A^ No 2 (68) HEPTAGONAL FUZZY NUMBER AND ITS APPLICATION__Volume 17, June 2022
Node7 and node8 are the predecessors of node 12.
E12 = max{E7 + t712,E8 + t812} = E7 + t712 = 19.6781 + 6.9246 = 26.6027
Iteration 12:
Node5 and node 9 are predecessor of node 13.
• E13 = max{E5 + t513,E9 + t913} = E9 + t913 = 22.6343 + 6.3420 = 28.9763
Iteration 13:
Node6, node10, node11, node12, node13 are the predecessor of node14.
•'• E14 = max{E6 + t614,E10 + ¡1014, En + tm4,E12 + ^1214,^13 + ¡1314} = E11 + t1114 = 37.9311 + 8.5898=46.5209
IV. Calculation of Latest times
In the above network, the end node is node14. Let E14 = L14 and label node14 as 46.5209. Iteration1:
The successor of node 13 is node14.
• L13 = L14 — t1314 = 46.5209 — 8.9228 = 37.5981
Iteration 2:
The successor of 12 is node14.
• Z12 = l14 — t1214 = 46.5209 — 5.0110 = 41.5099
Iteration 3:
The successor of 11 is node 14.
• L11 = L14 — ti114 = 46.5209 — 8.5898 = 37.9311
Iteration 4:
The successor of 10 is node14.
• L10 = l14 — t1014 = 46.5209 — 6.0092 = 40.5117
Iteration 5:
The successor of 9 is node13.
• l9 = L13 — t913 = 37.5981 — 6.3420 = 31.2561
Iteration 6:
The successor of node 8 is node 12.
• Z8 = L12 — t812 = 41.5099 — 10.0085 = 31.5014
Iteration 7:
The successor of node 7 is node11 and node 12.
• L7 = min{L11 — t711,L12 — t712} = L11 — t711 = 37.9311 — 18.2530 = 19.6781
Iteration 8:
The successor of node 6 is node 14.
• L6 = L14 — t614 = 46.5209 — 5.3437 = 41.1772
Iteration 9:
The successor of node5 is node 9 and node 13.
• L5 = min{L9 — t59,L13 — t513} = L9 — t59 = 31.2 5 61 — 12.00 46 = 19.25 1 5
Iteration 10:
The successor of node 4 is node8 and node9.
• L4 = min{L8 — t48,L9 — t49} = L9 — t49 = 31.2 5 61 — 4.92 77 = 26.32 84
Iteration 11:
The successor of node 3 is node7 and node 10.
•• L3 = min{L7 — t37,L10 — t310} = L7 — t37 = 19.6781 — 8.67 3 0 = 11.0051
Iteration 12:
The successor of node 2 is node7.
• L2=L7 — t27 = 19.6781 — 11.3 3 82 = 8.3 3 99
Iteration 13:
•'• Li = min[L2 - i12,L3 - t13,L4 - t14,L5 - t15,L6 - t16} = l2- t12 = 8.3399 - 8.3399 = 0
V. Calculation of Total float
Computed Earliest finish time, Latest start time and total float using formulas mentioned in procedure step 4, step 6, and step 7, respectively.
The earliest start and finish times, the latest start and finish times, total float of fuzzy activities are depicted in Table 5.
Table 5: The earliest, latest times and a total float of activities with defuzzified values
Activity ESy EF^ LSn LFij TF-1 rij
1^2 8.3399 0 8.3399 0 8.3399 0*
1^3 6.5085 0 6.5085 4.4966 11.0051 4.4966
1^4 5.1774 0 5.1774 21.151 26.3284 5.1774
1^5 10.9217 0 10.9217 8.3298 19.2515 7.6742
1^6 9.0061 0 9.0061 32.1711 41.1772 32.1711
2^7 11.3382 8.3399 19.6781 8.3399 19.6781 0*
3^7 8.6730 6.5085 15.1815 11.0051 19.6781 4.4966
3^10 9.5891 6.5085 16.0976 30.9226 40.5117 24.4141
4^8 3.4328 5.1774 8.6102 28.0686 31.5014 22.8912
4^9 4.9279 5.1774 10.1053 26.3282 31.2561 21.1508
5^9 12.0046 10.9217 22.9263 19.2515 31.2561 8.3298
5^13 19.0029 10.9217 29.9246 18.5952 37.5981 7.6735
6^14 5.3437 9.0061 14.3498 41.1772 46.5209 32.1711
7^11 18.2530 19.6781 37.9311 19.6781 37.9311 0*
7^12 6.9246 19.6781 26.6027 34.5853 41.5099 14.9072
8^12 10.0085 8.6102 18.6187 31.5014 41.5099 22.8912
9^13 6.342 22.6343 28.9763 31.2561 37.5981 8.6218
10^14 6.0092 16.0976 22.1068 40.5307 46.5399 24.4331
11^14 8.5898 37.9311 46.5209 37.9311 46.5209 0*
12^14 5.0110 26.6027 31.6137 41.5099 46.5209 14.9072
13^14 8.9228 28.9763 37.8991 37.5981 46.5209 8.6218
From the above table, the critical activities are 1^-2, 2^-7, 7^11, 11^14.
Therefore, the project critical path is 1 ^2^7^11 ^14.
As a result, the project will be completed in 46.5209= 46.5 days.
VI Results
Table 6 represents the fuzzy critical path and project completion time by proposal method and existing methods. The result graph is presented in Figure 6.
Table 6: Proposal method correlated with existing methods Ranking Method Critical Path Project completion time
Namarta (2017) 1 ^2^7^11 45.375
Hamildon (2021) 1 ^2^7^11 46.7853
Proposal method 1 ^2^7^11 46.5399
Project completion time
47 46,5 46 45,5 45 44,5
Namarta (2017) | Hamildon (2021) | Proposal ranking Figure 6: Proposal ranking results correlated with existing ranking results
VII Conclusion
This paper introduced a new ranking function in Heptagonal fuzzy number. The proposed ranking function is derived from the centroid of HFN. In the network, every activity period is expressed by an HFN. The duration of every activity is transformed into the normal number or crisp number by a new ranking function. This normal number is considered as the expected time of activity. A conventional procedure identified the fuzzy critical path and project completion time. Numerous experiments have been conducted, and the results are correlated with some of the available ranking formulas. The attained results are similar to existing ranking results and the same critical path in all the methods. The proposal ranking can also be applied to more complex project networks in the real world. We can apply the ranking function of HFN to solve game problems and transportation problems.
Acknowledgement
There is no conflict of interest from co-author.
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