PENTAGON FUZZY NUMBER AND ITS APPLICATION TO FIND FUZZY CRITICAL PATH

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1 Volume 114 No , ISSN: (printed version); ISSN: (on-line version) url: PENTAGON FUZZY NUMBER AND ITS APPLICATION TO FIND FUZZY CRITICAL PATH SyedaNaziyaIkram 1 and A.Rajkumar 1 Asst.Professor Department of Mathematics Hindustan Institute of Technology and Science Chennai syedai@hindustanuniv.ac.in Asst.Professor Department of Mathematics Hindustan Institute of Technology and Science Chennai arajkumar@hindustanuniv.ac.in February 8, 011 ijpam.eu Abstract The objective is to find the critical path for the Pentagon fuzzy number on the basis of α - cut interval by various methods. Considering the activity durations are pentagon fuzzy numbers, and we also calculate total float of each activity and its fuzzy critical path. Through an example, calculation steps in this method and the results are provided. Key Words and Phrases: Graph Networks, Pentagon fuzzy numbers, Triangular fuzzy numbers, α - cut Interval numbers, signed distance measures, Magnitude measure, Area measures, Critical path, Total float. 1 Introduction The concept of fuzzy set theory was introduced by Zadeh [1] in CPM techniques have become widely popular and it is used as valuable tools for the purpose of planning and scheduling complex projects. It provides an easy way to deal with the problems in which the source of Vagueness occurs and can be applied in various field such as artificial intelligence, control system, decision making, etc.,if the activity times in the project are familiar and known, CPM has been identified to be a useful method in an efficient way. To deal with Imprecise in real life situation many researchers used triangular and trapezoidal fuzzy numbers. An Another way to deal with such imprecise data is to employ 1 183

2 the fuzzy concept and its Application, where by the vague activity times can be represented by fuzzy numbers. Cheng proposed the coefficient of variance CV = α/µ, µ 0, α (i.e.),where standard error and - standard error and - mean. In particular, problems of determining possible values of latest start times and floats in network with imprecise activity duration which are represented by fuzzy or interval number have attracted many researchers. Abbasby and Asady suggested a sign distance method in 006. In this paper, section deals with the basic definitions. In section 3, shortest distance is calculated using pentagon fuzzy numbers by various methodologies. Basic definitions Definition 1. A Pentagon fuzzy number A p = (S 1, S, S 3, S 4, S 5 ) where S 1, S, S 3, S 4, S 5 are real numbers and its membership is given below µ A (x) = 0 if x S 1 1[x S 1] [S S 1] if S 1 x S 1 + 1[y S] [S 3 S ] if S x S 3 1 1[S4 y] [S 4 S 3] if S 3 x S 4 1[S 5 x] [S 5 S 4] if S 4 x S 5 0 if x S 5 Definition. (AlphaCut) The α-cut set is the set of elements whose membership degree in A p = (S 1, S, S 3, S 4, S 5 ) is defined as A p = x X/µ A (x) α ={[P 1 (α), P (α)],for α [0, 0.5]}{[Q 1 (α), Q (α)],for α [0.5, 1]} Hence, A α ={[α(s S 1 ) + S 1 ), α(s 5 S 4 ) + S 5 ] for α [0, 0.5]} {[α(s 3 S ) + S S 3, α(s 4 S 3 ) S 4 + S 3 ] for α [0.5, 1]} (1) Definition 3. -(Signed distance Measure) The signed distance measure for any closed interval [x,y] measured from 0 is defined as d([x, y], 0]= 1 (x+y). Definition 4. Let A P = (p, q, r, s, t) be a Pentagon fuzzy number such that p < q < r < s < t. It is converted in terms of triangular fuzzy number as A = (p, a 1 = (q+r+s) 3, t) such that p< a 1 < t Then,the magnitude measure of triangular fuzzy number A = (p, a 1, t) is defined as Mag (A) = A L(α)+A R(α)+a 1 αdα, α [0, 1] 1 0 Definition 5. Let A=(p,q,r,s,t ; λ) be a level λ pentagon fuzzy number such that p < q < r < s < t0 < λ 1. If r q p λ, then Area Measure =λ ((r q p+s+t)) and If r q p = λ, then Area measure = λ ((q p+λ+s+t r)). 184

3 3 Methods For Finding Fuzzy Critical Path (FCP) Problem Forward Pass Calculation:- First Calculating the Earliest Starting Time (EST) in the given Network. S j = Max[S i + t ij ] (3), i = number of preceding nodes t i j is the estimated duration of the activity i j. Earliest Finishing Time (EFT) = EST + Fuzzy activity time (4) Backward Pass Calculation:-Next, to calculate the Latest Finishing Time (LFT) in the given project Network R i = Min[R J t ij ] (5), j = number of succeeding nodes Latest Starting Time (LST) = LFT-Fuzzy activity Time (6) METHOD 1:(FCP method using Signed Distance Measure) Step 1: - Construct a network G (V, E), Where graph is made up of vertices (V) and edges(e). Here arc lengths (edge weight) are assuming as Pentagon Fuzzy number and converted into α cut interval number using Defn () Step : - Using equation (3), calculate Earliest Starting Time (EST). Step 3: - Using Equation (4), Calculate Earliest Finishing (EFT) Step 4: - Using equation (5), Calculate Latest Finishing Time (LFT) Step 5: -Using equation (6), Calculate Latest Starting Time (LST) Step 6: - Calculate Total Float (TF). Step 7: - Calculate Signed distance measure for each activity using Defn (3) Step 8: -If the value of signed distance measure is zero, then such activities are known as fuzzy critical activities and its corresponding path is called fuzzy critical path. Numerical Example: - 1. Assume a Network of Civil Project - Figure Results of the given network using signed distance measure is given in Table 3 185

4 METHOD : - (FCP method using Magnitude measure ) Step1: It is following as in procedure 1. Step : Assuming arc length as Pentagonal fuzzy numbers. Step 3: Expected time of Pentagonal Fuzzy number are converted in terms of triangular fuzzy number and then defuzzifying using definition (4) for each activity. Step 4: Then usual method is applied to find the (shortest distance )fuzzy critical path. Assuming the given network in Fig 1, and the arc length as Pentagonal Fuzzy number as same as given above (Refer Table 3). METHOD : (FCP method using Area Measure) Step 1: -Follows same as in procedure given above in (1) Step : -Now, the Expected time can be defuzzified using Defn (5)which is given interms of Pentagonal Fuzzy numbers and let (λ = 1) for all activities in the given network. Step 3: -Then usual methodis applied to identify the fuzzy critical path. (Refer table ) Activity Fuzzy Activity Time Fuzzy Activity Time TF Signed distance (Pentagonal (α-cut measure using Fuzzy Number) Interval number)α = 0.5 Defn (3) 1- (1,,3,4,5) (4,) (0,0) (6,7,8,9,10) (9,7) (5,5) 5-4 (11,1,13,14,15) (14,1) (15,15) (16,17,18,19,0) (19,17) (5,5) 5-5 (1,,3,4,5) (4,) (0,0) (6,7,8,9,30) (9,7) (-15,-15) (31,3,33,34,35) (34,3) (5,5) (36,37,38,39,40) (39,37) (0,0) (41,4,43,44,45) (44,4) (-15,15) -15 Table 1: Results of the Network Using Signed Distance Measure Activity Pentagonal Fuzzy Number Defuzzified Activity Time TF using Defn (5) 1- (16,17,18,19,0) (1,,3,4,5) (6,7,8,9,30) (31,3,33,34,35) (36,37,38,39,40) (41,4,43,44,45) Table : Result of the Network Using Area Measure Therefore, Path is the fuzzy critical path Hence is a fuzzy critical path 4 186

5 Activity Fuzzy Activity Time Fuzzy Activity Defuzzified Activity Time TF (Pentagonal Converted interms using Defn (4) Fuzzy Number) of triangular fuzzy number using Defn(4) 1- (1,,3,4,5) (1,3,5) ,7,8,9,10) (6,8,10) (11,1,13,14,15) (11,13,15) (16,17,18,19,0) (16,18,0) (1,,3,4,5) (1,3,5) (6,7,8,9,30) (6,8,30) (31,3,33,34,35) (31,33,35) (36,37,38,39,40) (36,38,40) (41,4,43,44,45) (41,43,45) Table 3: Result of the Network Using Magnitude Measure Conclusion In a real project network, fuzzy models plays a vital role in determining the critical path. Thus, in this paper some methods have been extended to find the fuzzy critical path in the weighted graph (Network) which helps us to decide the best possible critical path in fuzzy environment. The Pentagon Fuzzy number and its α-cut interval number using Signed distance measure, Area measure, and Magnitude measure has been discussed. It is easy to apply in the real life problems. References [1] M.Hapke and R. Slowinski, Fuzzy priority heuristics for project scheduling, Fuzzy sets and systems, vol. 83, no.3, pp.91-99,1996. [] J.S.Yao and F.T. Lin, Fuzzy Critical Path Method Based on Signed Distance ranking of fuzzy numbers, IEEE Transactions on systems, man and cybernetics-part A : systems and humans, Vol. 30, No.1., 000. [3] S.M. Chen and T.H. Chang, Finding Multiple Possible Critical Paths using Fuzzy PERT, IEEE Transactions on systems, man and cybernetics-part A : systems and humans, Vol. 31, No.6., 001. [4] G.J. Klir, B.Yuan, Fuzzy Sets and Fuzzy Logic : Theory and Applications, Prentice- Hall, International Inc., [5] S.Elizabeth,L.Sujatha.Critical Path problem under fuzzy Environment International Journal of Computer Applications( ) [6] R.Helen,G.Uma ANew Operation and Ranking on Pentagon Fuzzy Numbers, international Journal of Mathematical Sciences and application Vol.5,No (July-December 015) 5 187