Fuzzy Critical Path Method Based on a New Approach of Ranking Fuzzy Numbers using Centroid of Centroids

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1 16 International Journal of Fuzzy System Applications, 3(2), 16-31, April-June 2013 Fuzzy Critical Path Method Based on a New Approach of Ranking Fuzzy Numbers using Centroid of Centroids N. Ravi Shankar, GITAM University, Visakhapatnam, AP, India B. Pardha Saradhi, Dr. L.B. College, Visakhapatnam, AP, India S. Suresh Babu, GITAM University, Visakhapatnam, AP, India ABSTRACT The Critical Path Method (CPM) is useful for planning and control of complex projects. The CPM identifies the critical activities in the critical path of an activity network. The successful implementation of CPM requires the availability of clear determined time duration for each activity. However, in practical situations this requirement is usually hard to fulfil since many of activities will be executed for the first time. Hence, there is always uncertainty about the time durations of activities in the network planning. This has led to the development of fuzzy CPM. In this paper, a new approach of ranking fuzzy numbers using centroid of centroids of fuzzy numbers to its distance from original point is proposed. The proposed method can rank all types of fuzzy numbers including crisp numbers with different membership functions. The authors apply the proposed ranking method to develop a new fuzzy CPM. The proposed method is illustrated with an example. Keywords: Centroid of a Triangle, Critical Path Method, Fuzzy Numbers, Fuzzy Set Theory, Membership Function, Project Evaluation and Review Technique (PERT) INTRODUCTION Fuzzy set theory introduced for the first time in 1965 by Zadeh. Since then a lot of problems in fuzzy mathematics has been created and developed. In this highly competitive world, the socio-economic, business and financial scenarios are changing at a faster rate. An investor DOI: /fsa wants to optimize his resources for high profits and less risk and on-time delivery to customers. To maximize resource utilization and minimize overall cost, project management has always been an important issue for several agencies and industrial organizations. The network techniques used to handle project analysis are Project Evaluation and Review Technique (PERT) and Critical Path Method (CPM). A project, and more generally any activity network, is classically defined as a set of activities which must

2 International Journal of Fuzzy System Applications, 3(2), 16-31, April-June be performed according to some precedence constraints requiring that some activities cannot start before some others are completed. When resource constraints are not taken into account, a project can be represented by a directed acyclic graph where the nodes stand for activities and the arcs for precedence relations. In this context, the project manager generally aims at minimizing the completion time of the last task. When the activity times in a project are deterministic and known, CPM provides the minimal project duration and identifies the critical paths. Also, there are many cases where the activity times are not deterministic, but random assessments, and in this case PERT which is based on probability theory can be employed. However, in real world applications some activity times must be forecasted subjectively like using human judgement, decision-makers wisdom, professional knowledge, experience, instead of stochastic assumptions to determine activity times. An alternative way to deal with imprecise data is to employ the concept of fuzziness (Zadeh, 1965), whereby the vague activity times can be represented by fuzzy sets. In this paper a new method is proposed to find a critical path of a project network under fuzzy environment. First, we define a new ranking function based on centroid of centroids of generalized fuzzy numbers to find the maximum and minimum of two trapezoidal fuzzy numbers, whereas other methods used the ranking index proposed by Liou and Wang (1992) or Kaufmann and Gupta (1988). The proposed method is illustrated by a numerical example taken from existing method (Liang and Han, 2004) and it is shown that the results of the proposed method and the existing method are identical. The rest of the paper is organized as follows: some basic definitions, arithmetic operations of fuzzy sets available from existing literature are reviewed. Literature review of ranking fuzzy numbers is reviewed. Literature review of existing fuzzy critical path of a project network is reviewed. The proposed method of ranking fuzzy numbers using centroid of centroids is explained. New fuzzy critical path method is proposed by defining a fuzzy number using centroid of centroid. The proposed method is illustrated by a numerical example taken from existing method (Liang & Han, 2004) and it is shown that the results of the proposed method and the existing method are identical. Finally the conclusions are given. FUZZY BASIC DEFINITIONS In this section, some basic fuzzy basic definitions on fuzzy sets are presented (Kaufmann & Gupta, 1985): Definition 1: Let U be a Universe set. A fuzzy set à of U is defined by a membership function f : U [ 0, 1 ] where f is the à à grade of x in Ã, x U ; Definition 2: A fuzzy set à of universe set U is a fuzzy number if (i) à is normal i.e., Sup f ( x) = 1 and (ii) à is convex i.e.; x U A f ( λx + ( 1 λ) y) min ( f ( x), f ( y) ), x, y U A A A and: λ [ 0, 1 ] ; Definition 3: The membership function of the real fuzzy number à is given by: f A L f x a x b A ( ), <, w, b x c, ( x) = R f x c x d A ( ), <, 0, otherwise, where 0 < w 1 is a constant, a, b, c, d are L real numbers and f ( x) : [ a, b] [ 0, w], R f x c d w A ( ) : [, ] [ 0, ] are two strictly monotonic and continuous functions. It is standard A

3 18 International Journal of Fuzzy System Applications, 3(2), 16-31, April-June 2013 to write a fuzzy number as à = ( a, b, c, d ; w). If w = 1, then à = ( a, b, c, d ; 1 ) is a normalized fuzzy number, otherwise à is said to be a generalized or non-normal fuzzy number if 0 < w < 1 ; Definition 4: If the membership function is f ( x) piecewise linear, then à is said to à be a trapezoidal fuzzy number. The membership function of a trapezoidal fuzzy number is given by: w( x a), a x < b, b a w b x c f ( x ),, = à w( x d), c < x d, c d 0, otherwise, If w =1, then à = ( a, b, c, d ; 1 ) is a normalized trapezoidal fuzzy number, otherwise à is a generalized or non normal trapezoidal fuzzy number if 0 < w < 1. As a particular case if b = c, the trapezoidal fuzzy number reduces to a triangular fuzzy number given by à = ( a, b, d ; w ). If w =1, then à = ( a, b, d ) is a normalized triangular fuzzy number, otherwise à is a generalized or non normal triangular fuzzy number if 0 < w < 1. A Review of the Comparison between Fuzzy Numbers Ranking fuzzy numbers was first projected by Jain (1976) for decision making in fuzzy situations by indicating the vague quantity as a fuzzy set. Since then, different procedures to rank fuzzy quantities are proposed by various researchers. Bass and Kwakernaak (1977) projected a canonical way to expand the natural ordering of real numbers to fuzzy numbers. Baldwin and Guild (1979) compared fuzzy numbers on the same decision space and also pointed out that the method is having some troubling disadvantage. Yager (1980, 1981) introduced four indices to order fuzzy quantities in [0,1]. Kerre (1982) used fuzzy ranking in electro cardiological diagnostics and Dubois and Prade (1983) introduced a complete set of comparison indices in Zadeh s possibility theory. Chen (1985) presented ranking fuzzy numbers with maximizing set and minimizing set. Kolodziejczyk and Orlovsky (1986) and Nakamura (1986) presented the concept of decision making using fuzzy preference relations. Delgado et al. (1988) presented a procedure for ranking fuzzy numbers. Campos and Munoz (1989) presented a subjective approach for ranking fuzzy numbers. Kim and Park(1990) presented a method of ranking fuzzy numbers with index of optimism. Yuan (1991) presented a criterion for evaluating fuzzy ranking methods. Saade and Schwarzlander(1992) presented ordering fuzzy sets over the real line. Liou and Wang (1992) presented ranking fuzzy numbers with integral value. Choobineh and Li (1993) presented an index for ordering fuzzy numbers. Since then several methods have been proposed by various researchers which include ranking fuzzy numbers using area compensation (Fortemps & Roubens, 1996), distance method (Cheng, 1998) decomposition principle and signed distance (Yao & Wu, 2000). Wang and Kerre (2001) classified all the above ranking procedures into three classes. The first class consists of ranking procedures based on fuzzy mean and spread and second class consists ranking procedures based on fuzzy scoring, whereas the third class consists of methods based on preference relations and concluded that the ordering procedures associated with first class are relatively reasonable for the ordering of fuzzy numbers specially the ranking procedure presented by Adamo (1980) which satisfies all the reasonable properties for the ordering of fuzzy quantities. The methods presented in the second class are not doing well and the methods which belong to class three are reasonable. Later on, ranking fuzzy numbers by preference ratio (Modarres & Nezhad, 2001), left and right dominance (Chen and Lu, 2001), area between the centroid point

4 International Journal of Fuzzy System Applications, 3(2), 16-31, April-June and original point (Chu & Tsao, 2001), sign distance (Abbasbandy & Asady, 2006), distance minimization (Asady & Zendehnam, 2007) came into existence. Later Garcia and Lamata (2007), modified the index of Liou and Wang (1992) for ranking fuzzy numbers, by stating that the index of optimism is not alone sufficient to discriminate fuzzy numbers and proposed an index of modality to rank fuzzy numbers. Kumar et al. (2010) presented a procedure on ranking generalized trapezoidal fuzzy numbers based on rank, mode, divergence and spread. Rao and Shankar (2011) presented a method on ranking fuzzy numbers using circumcenter of centroids and index of modality. Most of the methods presented above cannot discriminate fuzzy numbers and some methods do not agree with human intuition whereas, some methods cannot rank crisp numbers which are a special case of fuzzy numbers LITERATURE REVIEW OF EXISTING FUZZY CRITICAL PATH METHODS Gazdik (1983) developed a fuzzy project network to estimate the activity durations and used fuzzy algebraic operations to calculate the critical path with project completion time. Kaufmann and Gupta (1988) dedicated a chapter to the fuzzy critical path method in which activity times are represented by triangular fuzzy numbers. A six step system is abridged for developing the activity estimates, determining float times for each activity, and identifying the fuzzy critical path in a fuzzy project network. Chang et al. (2002) combined the composite and comparison methods of analyzing fuzzy numbers into an efficient method for solving project scheduling problems. The comparison method first removes activities that are not highly critical paths. The composite method then determines the fuzzy path with the highest degree of criticality. The fuzzy Delphi method (Chang et al., 1995) is used to determine the fuzzy activity time estimates. The solution procedure is demonstrated in a 9 node, 14 activity project scheduling problem with activity times represented by triangular fuzzy numbers. Nasution (1994) argued that for a given α -cut level of the slack, the availability of the fuzzy slack in critical path models provides sufficient information to determine the critical path. A fuzzy procedure utilizing interactive fuzzy subtraction is used to compute the latest allowable and slack for activities. The procedure is demonstrated for a ten event network where activity times are represented by trapezoidal fuzzy numbers. Hapke et al. (1994) presented a fuzzy project scheduling decision support system. The fuzzy project scheduling system is used to allocate resources among dependent activities in a software project scheduling environment. The fuzzy project scheduling system used L-R type flat fuzzy numbers to model uncertain activity durations. Expected project completion time and maximum lateness are identified as the project performance measures and a sample problem is demonstrated for a software engineering project involving 53 activities. The fuzzy project scheduling system presented allows the estimation of project completion times and the ability to analyze the risk associated with overstepping the required project completion time. Lorterapong (1994) introduced a resource- constrainted project scheduling method that addresses three performance objectives: (1) expected project completion time; (2) resource utilization; and (3) resource interruption. Fuzzy set theory is used to model the vagueness that is inherent with linguistic descriptions often used by people when describing activity durations. The analysis presented provides a framework for allocating resources in an uncertain project environment. Chen et al. (1997) incorporated time-window constraint and time schedule constraint into the traditional activity network. They developed a linear time algorithm for finding the critical path in an activity network with these timeconstraints. Yao and Lin (2000) proposed a method for ranking fuzzy numbers without the need for any assumptions and used both positive and negative values to define ordering which

5 20 International Journal of Fuzzy System Applications, 3(2), 16-31, April-June 2013 then is applied to critical path method. They made a fuzzy activity network using triangular fuzzy numbers to represent the fuzzy data for the duration of each activity. Then they eliminated the fuzziness using the signed distance ranking for those fuzzy numbers to co-construct the activity network, in the fuzzy sense. After that the fuzzy critical path can easily be obtained. Chanas and Zielinski (2001) introduced the concept of criticality in the network with fuzzy activity times. They also presented two methods of calculation of the path degree of criticality. Lin (2001) presented a new approach to a fuzzy critical path method for activity network based on statistical confidence interval estimates and a ranking method for level (1-α) fuzzy numbers. His focus was to introduce an approach that combined fuzzy set theory with statistics that includes the signed distance ranking of level (1-α) fuzzy numbers derived from (1-α)*100%. Kuchta (2004) defined a fuzzy way of measuring the criticality of project activities and of the whole project. For this, he took into account the decision maker s attitude and project network structure. He argued that the criticality measurer obtained may serve as a measure of risk or of the supervision effort needed and can help in making the decision whether to accept or to reject the project. Chanas et al. (2002) introduced and analyzed the notion of the necessary criticality of a network with imprecisely defined by activity duration times. They also proposed algorithms of calculating the degree of the necessary criticality of paths. Lin (2002) introduced a new approach to fuzzy critical path method that based on statistical confidence interval estimates which is the extension of crisp activity network. He also focused to introduce the approach to combine statistics with fuzzy mathematics, which include the signed-distance ranking of level (1-α) fuzzy numbers. Chang (2002) proposed a method to systematically determine the critical path under the work flow model and give an overall example that shows how the method works. Lin and Yao (2003) proposed an approach that combines fuzzy mathematics with statistics to solve practical problems in unknown or vague situations. They introduced a fuzzy critical path method based on statistical confidence-interval estimates and a signed distance ranking for (1-α) fuzzy number levels. Dubois et al. (2003) presented the basis for a correct calculation of latest starting dates, slack times and criticality degrees of tasks in task network with fuzzy processing times. They showed that this problem is rather easy to solve for particular network topologies namely series-parallel graphs. Liang and Han (2004) presented an algorithm to perform fuzzy critical path analysis for project network problem. By using this algorithm, they showed that the ambiguities involved in the assessment activity times in a project network can be effectively improved and thus a more convincing and effective project management decision-making can be obtained. Zielinski (2005) extended some results for interval numbers to the fuzzy case for determining the possibility distributions describing latest starting times for activities. He also proposed the time algorithms for computing the intervals of the possible values of the latest starting times of an activity in general networks with interval durations and extended the results to the networks with fuzzy durations. Jassbi and Khanmohammadi (2005) introduced a new approach based on membership functions for estimated durations and the delays of activities. Also they showed that beta shape fuzzy values are used as the possibility of performing activities at estimated times. Han et al. (2006) used the trapezoidal fuzzy numbers to make the fuzzy measures of activity times characterized by linguistic value and proposed an algorithm for finding the fuzzy critical path of a project network. They also introduced the method that reduce the complexity of airport s ground operation model development and computations for solving problems, as well as incorporate the decision maker s risk attitude into the decision process. This method is utilized to perform critical path analysis for Chiang Kai- Shek airport s ground operation work. Tian and Li (2006) presented a method of modeling the well-define collaboration process based on workflow technology, probability

6 International Journal of Fuzzy System Applications, 3(2), 16-31, April-June theory and fuzzy theory. They also offered theoretic basis and available means of time management, resource scheduling and conflict solving in collaborative work process. Yakhchali et al. (2007a) focused on the full picture of necessarily critical path in the network with interval and fuzzy time lags. They generalized the results obtained for networks with interval activity duration times and time lags to the case of networks with fuzzy activity duration times. For this purpose, they implemented an algorithm for calculating the degree of necessary criticality of a path. They also proposed a linear programming approach to determine the necessity degree that a path is critical in the network with fuzzy time lags. Yakhchali et al. (2007b) assumed time lags which are common practice in the different projects, are imprecise and they discussed the problems of possibly critical paths in the networks with interval-valued activity and time lag durations. Chen (2007) proposed an approach to critical path analysis for a project network with activity times being fuzzy numbers, in that membership function of fuzzy total duration time is constructed which is based on the extension principle and linear programming. Koo et al. (2007) presented a formal identification and resequencing process to support the correct and rapid development of sequencing alternatives in construction schedules. Chen and Hsueh (2008) presented a simple approach to solve the critical path method problem with fuzzy activity times (being fuzzy numbers) on the basis of the linear programming formulation and the fuzzy number ranking method that are more realistic than crisp ones. They also defined that the most critical path and the relative path degree of criticality which are easy to use in practice. Yakhchali et al. (2008) introduced the algorithms for computing of the interval value of the latest starting times and maximal floats of activities in the network with interval activity and time lag durations. They also compared the complexity of proposed algorithm with former algorithm. They also generalized the interval activity and time lag durations into fuzzy numbers. Lin (2008) presented a fuzzy approach based on statistical confidence interval estimates and a distance ranking method for (1-α) fuzzy number levels. He also defined a theorem based on statistical confidence interval estimates through which the fuzzy critical path and fuzzy time-cost trade off problems can be solved efficiently. Sharafi et al. (2008) presented a new method based on fuzzy theory for solving fuzzy project scheduling in fuzzy environment. Also into this model, they assumed for first time that the relationship between activities are not crisp and are supposed fuzzy numbers. They have proposed a method that calculates all parameters of project such as earliest and latest start and finish time and slack times. Eshtehardian et al. (2008) introduced a new approach for scheduling problem, considering uncertainties in activities execution time by using fuzzy set theory. They showed that the project manager may apply his own risk acceptance level to obtain a critical path with new critical activities using α-cut property. They also demonstrated that the project manager could reflect his/her degree of optimism with choosing different value of optimism index. Yakhchali and Ghodsypour (2009) introduced the problems of determining possible values of earliest and latest starting times of an activity in networks with minimal time lags and imprecise durations that are rerpresented by means of interval or fuzzy numbers. Sireesha and Shankar (2010) presented a new method based on fuzzy theory for solving fuzzy project scheduling in fuzzy environment. Assuming that the duration of activities are triangular fuzzy numbers, they computed total float time of each activity and fuzzy critical path without computing forward and backward pass calculations. Shankar et al. (2010) presented an analytical method for measuring the criticality in a fuzzy project network, where the duration time of each activity is represented by a trapezoidal fuzzy number. They used a new defuzzification formula for trapezoidal fuzzy number and apply to the float time for each activity in the fuzzy project network to find the critical path. Sireesha et al. (2012) developed a new method to compute the fuzzy latest times and float times of activities for a project scheduling problem with LR fuzzy

7 22 International Journal of Fuzzy System Applications, 3(2), 16-31, April-June 2013 numbers as fuzzy activity times. Dehuri and Cho (2012) developed an algorithm for classification by learning fuzzy network with a sequence bound global particle swarm optimizer. A NEW APPROACH OF COMPARISON BETWEEN FUZZY NUMBERS USING CENTROID OF CENTROIDS Consider a generalized trapezoidal fuzzy number A = ( a, b, c, d; w) (Figure 1). The Centroids a + b + c w of the three triangles are G =,, c + b 2w G = 2c + d w, and G =, respectively. Equation of the line G 1 G 3 is w y = and G 3 2 does not lie on the line G 1 G 3. Therefore, G 1, G 2 and G 3 are non-collinear and they form a triangle. We define the centroid G ( x y ) of the à 0, 0 triangle with vertices G 1,G 2 and G 3 of the generalized trapezoidal fuzzy number A = ( a, b, c, d; w). as: G a 2b 5c d 4w ( x y,, ) = à 0 0 As a special case, for triangular fuzzy number A = ( a, b, d; w) i.e., c = b the centroid of Centroids is given by: G ( a b d w x, y ) 7, 4 = + + à R( A ) = x + y R( A 1 ) = ( a + 7b + c) + 16w We define ranking between fuzzy numbers as: let Ãi and Ãj be two fuzzy numbers, then: ( ) > ( ) then i. If R A R A A > A ; i j i j ii. If R( A R A i ) < ( j ) then A < A ; i j iii. If R( A R A i ) = ( j ) then in this case the discrimination of fuzzy numbers is not possible. Figure 1. Trapezoidal fuzzy number

8 International Journal of Fuzzy System Applications, 3(2), 16-31, April-June In this case we define the index a s s o c i a t e d w i t h t h e r a n k i n g a s I A βs A 1 β I A where α, β ( ) = ( ) + ( ) ( ) M β 0, 1 is the index of modality which represents the importance of central value against the extreme values x 0 and y 0, S A α M ( ) is the mode associated with the fuzzy number which is equal to b for a triangular fuzzy number A = ( a, b, d; w), and the average value of the area of stability for a trapezoidal fuzzy number and we define the index associated with the ranking as I A αy α x α ( ) = + ( 1 0 ) where 0 α 0, 1 is the index of optimism which represents the degree of optimism of a decision maker. The ranking has been done as follows: if I A I A A > A and if α, β ( i ) > α, β ( j ) then i j ( ) < ( ) then A > A. I A I A α, β i α, β j A NEW APPROACH OF FUZZY CRITICAL PATH METHOD USING CENTROID OF CENTROIDS A fuzzy project network G (V,E,T ) is a directed acyclic graph, where the vertices represent events and edges represent the activities to be performed in a project. Let V = {1, 2,, n} be a set of vertices, where 1 and n are initial and end events of the project, and each i belongs to some path from 1 to n. Let E = { a / a is an activity from i to j} be set of edges. For each activity a, a fuzzy activity t i j T is defined, where t is the time required for the completion of activity a. To define the fuzzy activity time, we estimate the crisp time t in network planning. This estimate is obtained by calculating the mean of the past statistical data of the concerned project. Let t be the time required for a population, and let t k be the time required for the past n statistical data, k = 1,2,, n, for each activity a. Let t 1 = n n k= 1 t k be a point estimate for the time required for a population t. Let T = { t a E} be a network based on statistical data. We calculate the variance of e a c h a c t i v i t y ( i, j ) u s i n g t h e n formula s = t t ( k ) where t n 1 k, k 1 k =1,2,,n are sample values corresponding to the network. Let 0 < α p < 1, p =1,2, α 1 + α 2 = α, 0 < α < 1 be level of significance for each activity a. The level (1- α) fuzzy number for the time required for each activity a is defined by: s s t = t t ( α ), t, t + t ( α ) n 1 1 n 1 2 n n and: 0 = ( 0, 0, 0) Distance from t distance ranking: to 0 is defined using our 2 s * 1 t = D( t ) = 9t + t ( α α ) + 16 n n Earliest, Latest times, and total float times are calculated using the formulae: E * * * * * * 1 = 0, E = max( E + t ), ES = E, j i Pj i i * * * = + * * * * * * * = E, L = min( L t ), LF = L, n n j i S i j j * * * = * * * * * * * * = = = j i EF ES t L LS LF t TF LF LS EF ES L E t.

9 24 International Journal of Fuzzy System Applications, 3(2), 16-31, April-June 2013 Figure 2. Activity network for the construction project

10 International Journal of Fuzzy System Applications, 3(2), 16-31, April-June Table 1. Activities in a construction project Activity Description Immediate Predecessor(s) A Obtain material for beams --- B Excavate foundations --- C Obtain bricks --- D Obtain wood --- E Obtain sanitary fittings, etc --- F Obtain electric equipment --- G Lay foundations A,B H Brick work C,G I Lay drains C,G J Place roof timbers D,H K Complete roofing J L Fit exterior doors J M Plumbing E,I,J N Electric wiring F,K O Plaster N,L,M P carpentry O Q Place sanitary fittings O R Fit doors, etc. J S Point brick work R,Q Critical activity of the network are the activities with zero total float time. Critical path of the network is obtained by considering the path containing critical activities. NUMERICAL EXAMPLE We constructed an activity network (Figure 2) for the construction project using Table 1. We considered 20 constructed projects for case study with the activity times and they are presented in Table 2. Confidence interval for each activity of the project network is calculated and presented in Table 3. New triangular fuzzy number is constructed and time characteristics of the activitiy network are calculated using distance measure on centroid of centroids. The calculated values are presented in Table 4. From Table 4, Critical path of the network is It is identical with the Liang and Han (2004) fuzzy critical path method. CONCLUSION AND DIRECTIONS FOR FUTURE RESEARCH In this paper a new method based on the fuzzy set theory has been developed to solve the project network problem under the fuzzy environment.

11 26 International Journal of Fuzzy System Applications, 3(2), 16-31, April-June 2013 Table 2. Activity times of construction projects

12 International Journal of Fuzzy System Applications, 3(2), 16-31, April-June Table 3. Confidence interval for each activity Activityi-j t s t s t α ν 1, α n s t t ν 2, n α s ν t t 1, n + α 1, ν s n A new approach of ranking fuzzy numbers using centroid of centroids of fuzzy numbers to its distance from original point has been proposed. The proposed ranking method has satisfied all the reasonable properties of ranking fuzzy numbers (Wang et al., 2001). We have applied the proposed ranking method to develop a new fuzzy CPM. It is shown that the results of the

13 28 International Journal of Fuzzy System Applications, 3(2), 16-31, April-June 2013 Table 4. Triangular fuzzy number and time characteristics Activity i-j t t * = d t, 0 ES * EF * LS * LF * TF * 1-2 (3.54, 4.08,4.80) (4.03,4.65,5.48) (21.18, 22.05, 23.21) (10.54,10.95, 11.51) (16.94, 17.5,18.26) (10.34, 10.7, 11.18) (0, 0, 0) (2.09,2.4,2.82) (6.34,6.65,7.07) (2.30, 2.58, 2.95) (2.20,2.45,2.79) (0, 0, 0) (3.15, 3.43, 3.80) (2.02,2.25,2.56) (4.08, 4.63,5.35) (4.33, 4.86, 5.61) (5.17, 5.6, 6.18) (5.01, 5.35, 5.80) (1.03, 1.2, 1.43) (1.191, 1.4,1.68) (8.39,8.6, 8.88) new fuzzy critical path method and the existing fuzzy critical method are identical. The proposed fuzzy critical path method is more efficient compare to other existing methods proposed so far. The proposed ranking function can be applied to fuzzy transportation problems, fuzzy assignment problems, fuzzy risk analysis problems and fuzzy travelling salesman problems.

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