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Adilakshmi Siripurapu, Ravi Shankar Nowpada RT&A, No 3 (69) CRITICAL PATH ANALYSIS BY PENTAGONAL FUUZY NUMBER_Volume 17, September 2022
Fuzzy Project Planning and Scheduling with Pentagonal
Fuzzy Number
Adilakshmi Siripurapu1, Ravi Shankar Nowpada2
Dept. of Basic Science and Humanities, Vignan's Institute of Information Technology (A), Duvvada, Visakhapatnam, AP, India1 slakshmij [email protected]
Dept. of Mathematics, Institute of Science, GITAM (Deemed to be University),
Visakhapatnam, AP, India2 Drravi68@gmail .com
Abstract
In optimization approaches such as assignment issues, transportation problems, project schedules, artificial intelligence, data analysis, network flow analysis, an uncertain environment in organizational economics, and so on, ranking fuzzy numbers is essential. This paper introduces a new fuzzy ranking in Pentagonal fuzzy numbers. Each activity's duration is expressed as a Pentagonal fuzzy number in the project schedule. The new ranking function transforms every Pentagonal fuzzy number into a crisp number (normal number). We calculated the fuzzy critical path using a new algorithm. These approaches are illustrated with a numerical example.
Keywords: Activity duration, centroid, fuzzy ranking, fuzzy critical path, fuzzy number, project schedule.
1. Introduction
The critical path approach is one of the most important concepts in network analysis. It is used to solve project challenges by creating networks and establishing each activity's earliest start and earliest finish date. It is also a scheduling algorithm for a collection of project networks. It is also generally associated mostly with Program Evaluation and Review Technique (PERT).
Zadeh [12] introduced the existence of 'fuzzy logic', which considers inaccuracies and inconsistencies. Several academics have used fuzzy numbers in various forms, such as fuzzy triangular numbers, Trapezoidal fuzzy numbers, etc.
In many practical situations, the variables that define information uncertainty or vagueness are usually triangular or trapezoidal fuzzy numbers. Lee et al. [5] introduced Pentagonal fuzzy numbers and generalized the results of addition, subtraction, multiplication, and division based on Zadeh's extension principle. Pathinadhan et al. [6] proposed a new form of the non-normal generalized pentagonal fuzzy number, and some of its arithmetic operations, centroid, and median were discussed. Siji et al. [8] solved network problems with Pentagonal Intuitionistic fuzzy numbers using the ranking approach. Arokiamary et al. [1] determined the critical path analysis in a project network using the fuzzy TOPSIS method. Uthra et al. [11] defined a Generalized Intuitionistic Pentagonal fuzzy number and developed a new ranking formula.
Sahaya Sudha et al. [7] solved the fuzzy linear programming problem by applying the pentagonal fuzzy ranking function. Uma Maheswari et al. [10] introduced a simple approach for solving fuzzy transportation problems using fuzzy pentagonal numbers. Avishek Chakraborty et al. [2] studied interval-valued pentagonal fuzzy numbers, properties, ranking, and defuzzification. They solved the game problem by using a ranking function in fuzzy pentagonal numbers. Someshwar et al. [9] solved the linear programming problem using the pentagonal fuzzy ranking function. Das et al. [3] suggested a novel pentagonal neutrosophic technique for solving the linear programming problem.
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 nA(x), which associates with each point inX a real number in the interval [0,1], with the value of nA(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, pA(x) = 1 or 0 according to x does or does not belong to A.
2.2 Fuzzy Number [4]
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.
2.3 Pentagonal Fuzzy Number (PFN) [6]
A pentagonal fuzzy number Ap = (p#, p$,p%,p&, Ps) where p#, P$,P%,P&, Ps real numbers and its membership function is defined by:
2. Basic Definitions
In this section we look at a few definitions.
2.1 Fuzzy Set [12]
V
0; otherwiise
The Pentagonal Fuzzy Number diagram is represented in Figure 1.
0.5
Pi Vi Ps P4 Ps
Figure 1: Pentagonal Fuzzy Number
2.4 Generalized Pentagonal Fuzzy Number (GPFN) [6]
The generalized pentagonal fuzzy number Äp = (p#, p2, p3, p4, p5; m) its membership function is expressed as;
This section, proposed a new ranking function in Pentagonal fuzzy number.
Divide the pentagon into two triangles and one rectangle. Let Gi, G2, and G3 be the centroid of the three plane figures, respectively. The centroid of a fuzzy pentagon number is supposed to indicate the pentagon's balancing point (Figure 3).
The Proposal ranking in the Pentagonal fuzzy number diagram is represented in Figure 3.
V 0; otherwiise
The generalized Pentagonal Fuzzy Number diagram is represented in Figure 2.
Pi Pj Pj P. P' Figure 2: Generalized Pentagonal Fuzzy Number
3. Methodology
(p„o>)
pi pi pi Pa p>
Figure 3: Proposed Ranking Approach
To determine the ranking of a generalized pentagon fuzzy number, the in centre of the centroid Gi, G2, G3 is used as a point of reference.
Consider the generalized pentagon fuzzy number Ap = (p#, p2, p3, p4, p5, w)
The three plane figures' centroid isG# = (E*+S+£", *), G$ = ("$+$"+"%, *), G% = , *)
respectively.
The centroid of G#,G2 and G3 is , g).
N (4P# + 7 P2 + 14p3 + 7p4 + 4P5 \ % (x0,y0) = ^-—-,5^/36j
The in centre of the centroid with Euclidean distance is;
J*. + y0$
Consider that the centroid's center with Euclidean distance is a new ranking function in the generalized Pentagonal fuzzy number.
Therefore, a new ranking in the Generalized Pentagonal fuzzy number is;
= J*. + y<2
3.1 Fuzzy Critical Path Analysis
The primary objective of the fuzzy critical path is to estimate the total project duration and assign start and finish dates to all project activities. This makes it possible to compare the actual progress to the estimated duration.
The following fuzzy factors should be known to prepare the project schedule.
(i) Project completion time
(ii) Each activity's earliest and latest times
(iii) Critical activities and the critical path
(iv) Float for each activity (i.e., the time required to complete a non-critical activity can be delayed without affecting the overall project completion time)
3.1.1 Notations
FES ¿J = Earliest start of time of an activity (i,y)
Adilakshmi Siripurapu, Ravi Shankar Nowpada RT&A, No 3 (69)
CRITICAL PATH ANALYSIS BY PENTAGONAL FUUZY NUMBER_Volume 17, September 2022
FLSij = Latest start time of an activity (i,y) FEFij = Earliest finish time of an activity (i,y) FLFij = Latest finish time of an activity(ij) Ftij = Total duration of an activity (ij) FTFij =Total float of an activity (i,y)
3.1.2 Algorithm for Fuzzy Critical Path
Stepl: Construct a fuzzy project network with predecessor and successor events. Step2: Express every activity time as PFN.
Step3: Transformed every PFN as a crisp number using a new ranking function.
Step4: Calculate earliest start time, FF^j = max^^S^ + tij}, i = number of preceding nodes.
Step5: Calculate earliest finish time, FFFjj = FF.^ + tij.
Step6: Calculate latest finish time FLFij = min{FL^jj — tij}, j = number of succeeding nodes
Step7: Calculate latest start time, FLS^- = FLFij — tij
Step8: Calculate Total float, FTF^- = FLFjy — FFF^- or FLS^ — FF.Sij
4 Application of Pentagonal fuzzy numbers in a project schedule
Consider the following fuzzy project network, in which the Pentagonal fuzzy number represents each activity. The fuzzy project network has seven nodes and nine activities. This example shows how to schedule a construction project using a project Network. My objective is to analyze the maximum path that is the essential Critical path for the construction process.
Table 1 represents the activities, their description and duration periods. Figure 4 represents the project network diagram.
Table 1: Project Network Description
Activity Pentagonal fuzzy numbers
1^2 (1,2,3,4,5)
1^3 (6,7,8,9,10)
2^4 (11,12,13,14,15)
3^4 (16,17,18,19,20)
2^5 (21,22,23,24,25)
3^6 (26,27,28,29,30)
4^7 (31,32,33,34,35)
5^7 (36,37,38,39,40)
6^7 (41,42,43,44,45)
Figure 4: Project Network
4.1 Expected time of activities
Pentagonal fuzzy number transformed into activity duration by proposal ranking function. This activity duration is taken as the time between the nodes, and the fuzzy critical path is calculated by applying an algorithm. The expected time of activities is represented in Table 2, and the related diagram is represented in Figure 5.
Table 2: Expected time of activities
Activity i —> j Pentagonal fuzzy number Expected time
(1,2,3,4,5) 3.0032
1^3 (6,7,8,9,10) 8.0012
2^4 (11,12,13,14,15) 13.0007
3^4 (16,17,18,19,20) 18.0005
2^5 (21,22,23,24,25) 23.0004
3^6 (26,27,28,29,30) 28.0003
4^7 (31,32,33,34,35) 33.0002
5^7 (36,37,38,39,40) 38.0002
6^7 (41,42,43,44,45) 43.0002
28.0003
Figure 5: Expected time of Activities
Adilakshmi Siripurapu, Ravi Shankar Nowpada RT&A, No 3 (69) CRITICAL PATH ANALYSIS BY PENTAGONAL FUUZY NUMBER_Volume 17, September 2022
4.2 Earliest, latest times and Total float of fuzzy activities
Computed Earliest, latest times and total float using formulas mentioned in procedure step 4, step 5, step6, step7 and step 8, respectively.
The earliest, latest and total float times of fuzzy activities represented in Table 3.
Table 3: The earliest, latest times and total float of fuzzy activities with defuzzified values of PFN i ^ j Fty FESij FEFi0 FLSij FLFi0 F7Ty
1^2 3.0032 0 3.0032 14.9979 18.0011 14.9979
1^3 8.0012 0 8.0012 0 8.0012 0*
2^4 13.0007 3.0032 16.0039 33.0008 46.0015 29.9976
3^4 18.0005 8.0012 26.0017 28.001 46.0015 19.9998
2^5 23.0004 3.0032 26.0036 18.0011 41.0015 14.9979
3^6 28.0003 8.0012 36.0015 8.0012 36.0015 0*
4^7 33.0002 26.0017 59.0019 46.0015 79.0017 19.9998
5^7 38.0002 26.0036 64.0038 41.0015 79.0017 14.9979
6^7 43.0002 36.0015 79.0017 36.0015 79.0017 0*
4.3 Results
According to the fuzzy total float, the fuzzy critical activities are 1^3, 3^6, 6^7.
Therefore, the critical path of the fuzzy project network is
Hence, the total duration of the project network is 79.0017=79 days.
Figure 6: Fuzzy Critical Path
5. Conclusion
This paper introduced a new ranking function in Pentagonal fuzzy numbers. The proposed ranking function is derived from the centroid of PFN. In the network, every activity period is expressed by a PFN. The duration of every activity is transformed into the normal number or crisp number by a new ranking function. This normal number is considered the expected time of activity. The fuzzy critical path algorithm was used to identify the fuzzy critical path and project completion time. The proposal ranking can also be applied to more complex project networks in the real world. We can apply the ranking function of PFN to solve game problems and transportation problems.
References
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