A NEW APPROACH FOR FUZZY CRITICAL PATH METHOD USING OCTAGONAL FUZZY NUMBERS

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1 Volume 119 No , ISSN: (on-line version) url: ijpam.eu A NEW APPROACH FOR FUZZY CRITICAL PATH METHOD USING OCTAGONAL FUZZY NUMBERS D. STEPHEN DINAGAR 1 AND N. RAMESHAN 2 1 ASSOCIATE PROFESSOR, PG AND RESEARCH DEPARTMENT OF MATHEMATICS, TBML COLLEGE, PORAYAR , INDIA. DSDINA@REDIFFMAIL.COM 2 ASSISTANT PROFESSOR, DEPT. OF MATHEMATICS, SRMIST, VADAPALANI, CHENNAI , INDIA. NRAMESHAN14@GMAIL.COM Abstract. This paper presents a new approach for finding the solution of fuzzy criticality by using octagonal fuzzy number in which the duration of all fuzzy activities are given. This analysis given that fuzzy number is used in this research article is more supportive for finding the critical path in the fuzzy project network. A numerical examples to compare our proposed method with triangular and trapezoidal fuzzy numbers. Based on this method management decision-making can be obtained effectively Key Words: Fuzzy Numbers, Octagonal fuzzy number, Project network, start, finish, Critical path. 1. Introduction In many complicated projects, CPM (critical path method) is act an important role in planning, scheduling and control over the projects [1]. Basirzadeh et al [4] have given a method for ranking fuzzy number using α-cuts. In this propose method, we are using arithmetic operations and ranking method for solving fuzzy project scheduling problems. This paper presents another approach to analyze the critical path in a project network with octagonal fuzzy numbers. A simple method that is simple to value and improves difficulty for the problem of computing latest starting times of activities in networks using fuzzy intervals with fuzzy activity durations Key words and phrases. Fuzzy Numbers, Octagonal fuzzy number, Project network, start, finish, Critical path

2 2 D. STEPHEN DINAGAR AND N. RAMESHAN is developed [6]. M.Shanmugasundari and K.Ganesan proposed a new approach for finding fuzzy floats and fuzzy critical path problems[5]. S.Narayanamoorthy and S.Maheswari analyzed the critical path method where the processing times of all activities follow octagonal fuzzy numbers. The paper is organized as follows: In Section 2, some fundamental concepts, definitions are reclaimed. In Section 3, Arithmetic operations of octagonal fuzzy numbers and the ranking method is proposed. In Section 4, Algorithm of fuzzy critical path is given. In Section 5, a numerical example is solved based on ranking method. Finally the paper completes with a conclusion. 2. Preliminaries In this section, we represent basic idea about the fuzzy numbers and definitions which involved in the research work. Fuzzy set theory have been defined by Zadeh [2] and the fundamental concepts are given by Dubois and Prade are applied to fuzzy environment [3] Definitions Definitions 1: (Fuzzy set). If X is a collection well defined objects (say x) then the fuzzy set A can be defined as à = {(x : µ x)/x X}, where µ x : R [0, 1] and is upper semi continuous. µ x is called the membership function of fuzzy set. The membership function maps each element of X to a membership value between 0 and Definition 2: (Fuzzy Number). The fuzzy number A is a fuzzy set whose membership function satisfies the following conditions: (1) µ A (x) is piecewise continuous (2) A fuzzy set A of the universe of discourse X is convex. (3) A fuzzy set of the universe of discourse X is called a normal fuzzy set if x i X exists Definition 3: (Generalized fuzzy number). A fuzzy set A is defined on universal set of real numbers is said to be a generalized fuzzy number if its membership function has the following conditions: (1) µ A (x) : R [0, 1] is continuous (2) µ A (x) = 0 for all x A (, a] [d, ) 358

3 A NEW APPROACH FOR FCPM USING OFN 3 (3) µ A (x) is strictly increasing on [a,b] and strictly decreasing on [c,d] (4) µ A (x) = w for all x [b, c], where 0 < w Definition 4:- (Octagonal fuzzy number). A fuzzy number (A,B,C) is normal octagonal fuzzy number expressed by (A, B, C) = (a, b, c, d, e, f, g, h), where (a, b, c, d, e, f, g, h) are real numbers and its membership functions µ A (x), µ B (x) and µ C (x) are given below µ A (x) = a x d 1, b x c x h, e x h e h 0, otherwise x a, d a µ A (x) = µ B (x) = a x b 0.6, b x g x g, g x h g h 0, otherwise x a, b a a x c 0.8, c x f x h, f x h f h 0, otherwise x a, c a 3. Arithmetic Operation on Octagonal fuzzy numbers Let A 1 = (a 1, a 2, a 3, a 4, a 5, a 6, a 7, a 8 ) and A 2 = (b 1, b 2, b 3, b 4, b 5, b 6, b 7, b 8 ) be two octagonal fuzzy numbers respectively Fuzzy numbers addition. : (a 1, a 2, a 3, a 4, a 5, a 6, a 7, a 8 ) (b1, b 2, b 3, b 4, b 5, b 6, b 7, b 8 ) = (a 1 + b 1, a 2 + b 2, a 3 + b 3, a 4 + b 4, a 5 + b 5, a 6 + b 6, a 7 + b 7, a 8 + b 8 ) Fuzzy numbers subtraction. : (a 1, a 2, a 3, a 4, a 5, a 6, a 7, a 8 ) (b 1, b 2, b 3, b 4, b 5, b 6, b 7, b 8 ) = [max(0, a 1 b 8 ), max(0, a 2 b 7 ),, max(0, a 3 b 6 )max(0, a 4 b 5, max(0, a 5 b 4 ), max(0, a 6 b 3 ), max(0, a 7 b 2 ), max(0, a 8 b 1 )] Fuzzy Number Scalar Multiplication: KǍ = K (a 1, a 2, a 3, a 4, a 5, a 6, a 7, a 8 ) = (Ka 1, Ka 2, Ka 3, Ka 4, Ka 5, Ka 6, Ka 7, Ka 8 )., K 0 359

4 4 D. STEPHEN DINAGAR AND N. RAMESHAN Ranking of Trapezoidal Numbers: For Comparing fuzzy numbers, The parametric methods are more efficient approach by use of a ranking function R : F (R) R which associated defuzzication of every fuzzy number. For octagonal fuzzy number Ǎ = (a 1, a 2, a 3, a 4, a 5, a 6, a 7, a 8 ) F (R), Ranking function R : F (R) R defined as R ( Ǎ ) = ( a 1+a 2+a 3+a 4+a 5+a 6+a 7+a 8 8 Let Ǎ1 = (a 1, a 2, a 3, a 4, a 5, a 6, a 7, a 8 ) F (R) and Ǎ 2 = (b 1, b 2, b 3, b 4, b 5, b 6, b 7, b 8 ) F (R) be any two octagonal fuzzy numbers. By the comparison, (1) Ǎ1 Ǎ2 R(Ǎ1 ) R(Ǎ2) (2) Ǎ1 Ǎ2 R(Ǎ1 ) R(Ǎ2) (3) Ǎ1 Ǎ2 R(Ǎ1 ) R(Ǎ2) (4) Ǎ1 Ǎ2 0 R(Ǎ1 ) R(Ǎ2) = 0 ) 3.1. Algorithm: (Fuzzy Critical Path). Step 1: To identify to all fuzzy activities with respective duration Step 2: To find the precedence relationships of each activities by applying fuzzy ranking function. Step 3: To construct project network diagram Step 4: To compute earliest start time (ES) ij which is the earliest occurrence time of zero for the initial event and earliest finish time(ef) ij which is the earliest starting time + the activity time. ie, (Ef) ij = (ES) ij +D ij, where D ij is duration of activity. event time = max{ (ES) ij + D ij } Step 5: Calculate the latest occurrence time of event i (i < j). This is the minimum of the latest start times(ls) ij of all the activities from the event. event time = min{ (LF ) ij D ij }, where D ij is duration of activity. 4. Numerical Example Let V = {1, 2, 3, 4, 5, 6, 7, 8} be the set of fuzzy events, fuzzy activity time for each activity is shown in table (durations in hours) Table 1: (Activity duration of each activity) 360

5 A NEW APPROACH FOR FCPM USING OFN 5 Figure 1. Network Diagram Activity Duration 1-2 0,2,3,4,5,6,7, ,4,5,6,7,7,8, ,4,4,5,6,6,7, ,5,5,6,7,7,9, ,7,7,8,9,10,11, ,9,10,11,12,14,15, ,7,8,9,10,11,12, ,12,14,15,16,18, ,12,14,15,15,16,18, ,13,15,16,17,18,19,20 To convert the given duration fuzzy time à = (a 1, a 2, a 3, a 4, a 5, a 6, a 7, a 8 ) into the form of à = ([a 5, a 4 ], α 1 = a 4 a 3, α 2 = a 6 a 5, α 3 = a 3 a 2, α 4 = a 7 a 6, ) (α 5 = a 2 a 1, α 6 = a 8 a 7 ) 5. Conclusion The critical path is This method is proposed to find latest fuzzy time, fuzzy float times of activities in a fuzzy project network with octagonal fuzzy interval times. This shows that there is an alternative procedure for finding the critical path using octagonal fuzzy numbers. This way is to be identified that the result is better than the other existing methods. 361

6 6 D. STEPHEN DINAGAR AND N. RAMESHAN Table 2: Total Float-Calculation Activity Duration (0,2,3,4,5,6,7,7) (4,4,5,6,7,7,8,9) (3,4,4,5,6,6,7,8) (0,5,5,6,7,7,9,10) Fuzzy duration Total Float ([5,4],1,1,1,2,0) ([7,6],1,0,1,1,0,1) ([6,5],1,0,0,1,1,1) ([7,6],1,0,0,2,5,1) ([0,0],0,0,0,0,0,0) ([0,0],0,0,0,0,0,0) ([0,0],0,0,0,0,0,0) ([5,4],1,1,1,1,2,0) ([5,4],1,1,1,2,0) ([7,6],1,0,1,1,0,1) ([6,5],1,0,0,1,1,1) ([12,10],2,1,1,3,7,1) ([7,4],3,2,4,3,1,3) ([3,-3],3,2,4,3,1,3) ([15,10],3,2,4,3,1,3) ([11,9],3,2,4,3,1,3) ([11,9],3,2,4,3,1,3) ([9,4],3,2,4,3,1,3) ([20,16],3,2,4,3,1,3) ([19,16],3,2,4,3,1,3) ([7,4],3,2,4,3,1,3) ([3,-3],3,2,4,3,1,3) ([15,10],3,2,4,3,1,3) ([7,4],3,2,4,3,1,3) Activity Duration (6,7,7,8,9,10,11,12) (8,9,10,11,12,14,15,16) (6,7,8,9,10,11,12,13) Fuzzy duration Total Float ([9,8],1,1,0,1,1,1) ([12,11],1,2,1,1,1,1) ([10,9],1,1,1,1,1,1) ([5,4],1,1,1,1,2,0) ([7,6],1,0,1,1,0,1) ([7,6],1,0,1,1,0,1) ([14,12],2,2,1,2,3,1) ([19,17],2,2,2,2,1,2) ([17,15],2,1,2,1,0,1) ([12,7],3,2,4,3,1,3) ([9,4],3,2,4,3,1,3) ([10,4],3,2,4,3,1,3) ([20,16],3,2,4,3,1,3) ([20,16],3,2,4,3,1,3) ([19,14],3,2,4,3,1,3) ([8,2],3,2,4,3,1,3) ([3,-3],3,2,4,3,1,3) ([4,-3],3,2,4,3,1,3) Activity Duration (10,12,14,15,15,16,18,20) (12,12,14,15,15,16,18,20) 13,13,15,16,17,18,19,20 Fuzzy duration Total Float ([15,5],1,1,2,2,2,2) ([15,4],1,1,2,2,0,2) ([17,16],1,1,2,2,0,2) ([12,10],2,1,1,3,7,1) ([19,17],2,2,2,2,1,2) 10,11,13,15,17,18,20,21 ([27,25],3,2,3,5,9,3) ([34,31],3,3,4,4,1,4) ([10,4],3,2,4,3,1,3) ([19,16],3,2,4,3,1,3) ([20,16],3,2,4,3,1,3) ([19,14],3,2,4,3,1,3) ([34,31],3,2,4,3,1,3) ([34,31],3,2,4,3,1,3) ([34,31],3,2,4,3,1,3) ([10,4],3,2,4,3,1,3) ([3,-3],3,2,4,3,1,3) ([4,-3],3,2,4,3,1,3) 362

7 A NEW APPROACH FOR FCPM USING OFN 7 References [1] A.I. Slyeptsov, T.A. Tyshchuk, Fuzzy temporal characteristics of operations for project management on the network, European Journal of Operational Research,147 (2003), [2] Zadeh,L.A., Fuzzy sets, Information and control. Vol.8, pp , [3] Dubois.D and Prade.H., Operations in fuzzy numbers, The International Journal of Systems Sciences, Vol.9, pp ,1978. [4] H.Basirzadeh, R.Abbasi, A new approach for ranking fuzzy numbers based on α-cuts, JAMI, Journal of Applied Mathematics, & Informatic, 26(2008), [5] M.Shanmugasundari and K.Ganesan, Project Scheduling Problems Under Fuzzy Environment: A New Solution Approach, International Journal of Pure and Applied Mathematics, Vol.95, No , [6] N.Ravi Shankar, V.Sireesha and P.Phani Bushan Rao, Critical path analysis in the fuzzy project network, Advances in Fuzzy Mathematics, Vol.5, no.3, pp , [7] S.Narayanamoorthy and S.M.Maheswari, The intelligence of octagonal Fuzzy number to determine the fuzzy critical path A new ranking method. Hindwai Scientific programming Vol

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