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A New Approach to find Total Float time and

Critical Path in a fuzzy Project Network

V. Sireesha , N. Ravi Shankar

Dept. of Applied Mathematics, GIS, GITAM University, Visakhapatnam, India

Abstract

The purpose of the critical path method (CPM) is to identify the critical activities in the critical path of an activity network. In the real world for many projects we have to use human judgment for estimating the duration of activities. However, the unknowns or vagueness about the time duration for activities in network planning, has led to the development of fuzzy CPM. A way to deal with this imprecise data is to employ the concept of fuzziness, where the vague activity times can be represented by fuzzy sets. In this paper a new method based on fuzzy theory is developed to solve the project scheduling problem under fuzzy environment. Assuming that the duration of activities are triangular fuzzy numbers, in this method we compute total float time of each activity and fuzzy critical path without computing forward and backward pass calculations. Through a numerical example, calculation steps in this method and the results are illustrated. Compare with other fuzzy critical method the proposed method is simple, fast and effective to find total float time of each activity and fuzzy critical path in a fuzzy project network.

Keywords : Fuzzy sets , Triangular fuzzy numbers, total float, critical path.

1.Introduction

Since the late 1950s, Critical Path Method (CPM) techniques have become widely recognized as valuable tools for the planning and scheduling complex projects. When the activity times in the project are deterministic and known, CPM has been demonstrated to be a useful method in managing projects in an efficient manner to meet the challenge [1]. There are many cases where the activity times may not be presented in a precise manner. To deal quantitatively with imprecise data, the program evaluation and review technique (PERT) [1,2] and Monte Carlo simulation [3] based on the probability theory can be employed. In real world applications some activity times must be forecasted subjectively; for example, we have to use human judgment instead of stochastic assumptions to determine activity times. An alternative way to deal with imprecise data is to employ the concept of fuzziness [4], whereby the vague activity times can be represented by fuzzy numbers. Fuzzy numbers are used to describe uncertain activity durations, reflecting vagueness, imprecision and subjectivity in the estimation of them. The main advantages of methodologies based on fuzzy theory are that they do not require prior predictable regularities or posterior frequency distributions, and they can deal with imprecise input information containing feelings and emotions quantified based on the decision-maker’s subjective judgment.

There have been several attempts in the literature to apply fuzzy numbers to the critical path method since the late 1970s and it has led to the development of fuzzy CPM [5-12].

In particular, problems of determining possible values of latest starting times and floats in networks with imprecise activity durations which are represented by fuzzy or interval numbers have attracted many researchers [13-17]. These methods compute the possible values of the earliest starting times by means of a forward recursion procedure comparable to the one used in the traditional CPM. The backward recursion takes the imprecion of some durations into account twice so the backward recusion does not work [18]. Kaufmann and Gupta [19], Hapke et al. [20] , Hapke and Slowinski [21] and Rommelfanger [22] proposed a backward recursion that relies on the optimistic fuzzy subtraction and they provided good results for particular networks but these methods fail to compute the fuzzy latest starting times and floats in general networks. Nasution [23] resorts to symbolic computations on variable duration times. McCahon and Lee [24], Mon et al. [25] and Yao and Lin [26] proposed to go back to classical CPM via defuzzification of the fuzzy durations. McCahon [27] proposed to compute approximated fuzzy floats of activities from the fuzzy starting times obtained by the forward and backward recursions.

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have provided a complete solution to the problem of finding the maximal float of an activity and Yakhchali and Ghodsypour [32] have proposed a hybrid genetic algorithm for the problem of finding the minimal float of an activity.

Chanas and colleagues proposed a series studies on the topic of the fuzzy project scheduling. For example, Chanas et al. [33] studied necessarily critical activities; Chanas and Zielinski [34,35] discussed the complexity of criticality; Chanas and Zielinski [36] proposed two methods of calculating the degree of possible criticality of paths. Problems related to necessarily and possibly critical paths in networks with imprecise activity and time lag durations have been discussed by Yakhchali et al. [37,29] . Chen and Hsueh [38] and Chen [39] proposed an approach based on the extension principle and linear programming formulation to critical path analysis in networks with fuzzy activity durations. Chen and Huang [40] combined fuzzy set theory with the traditional methods to compute the critical degrees of activities an paths. Instead of the traditional objective, minimizing the makespan of the project, Yakhchali and Ghodsypour [37] discussed project scheduling problem with irregular starting time costs in networks with imprecised durations.

In this paper, we propose a fuzzy critical path method to compute total float time of each activity and fuzzy critical path in a fuzzy project network without computing forward and backward pass calculations.

2. The Fuzzy critical Path method

2.1 Fuzzy Sets and Arithmetic Operations

Classical sets contain objects that satisfy precise properties of membership; fuzzy sets contain objects that satisfy imprecise properties of membership, i.e., membership of an object in a fuzzy set can be approximate.

Example : The set of heights from 5 to 7 feet is precise (crisp). The set of heights in the region around 6 feet is imprecise (fuzzy).

For crisp sets an element x in the universe X is either a member of some crisp set A or not. Mathematically, the membership function (binary membership) can be represented as

A

x

A

x

x

A

,

0

,

1

)

(

Zadeh extended the notion of membership to accommodate various “degrees of membership” on the real continuous interval [0,1], where the endpoints of 0 and 1 conform to no membership and full membership, respectively, just as the membership function does for crisp sets, but where the infinite number of values in between the endpoints can represent various degrees of membership for an element x in some set on the universe.

The sets on the universe X that can accommodate “degrees of membership” were termed by Zadeh as “fuzzy sets.” Example : Consider a set A consisting of heights near 6 feet. There is not a unique membership function A for A.

The analyst must decide the membership function . Plausible properties of this membership function might be (1) Normality (A(6) = 1),

(2) Monotonicity ( the closer A is to 6, the closer A is to 1), and

(3) Symmetry (numbers equidistant from 6 should have the same value of A).

A fuzzy number ~

A

is a fuzzy set of the real line R which is continuous, i.e., its membership function ~ A

is continuous normalized, i.e.,  x , ~

A

(x) =1 and convex, i.e.,  x, y  R ,  z [x,y] , ~ A

(z)  min ( ~ A

(x) ,

~ A

(y) ).

A fuzzy set ~

A

is called positive if its membership function is such that ~ A

(x) = 0  x  0.

Trapezoidal Fuzzy number is a convex fuzzy set which is defined as ~

A

= (x, ~ A

(3)

d

x

d

x

c

d

c

d

x

c

x

b

b

x

a

a

b

a

x

a

x

x

A

0

1

0

)

(

~

Let 1 ~

A

and 2 ~

A

be two trapezoidal fuzzy numbers parameterized by the quadruple (

a

1,b1,c1,d1) and (

a

2,b2,c2,d2),

respectively. The simplified fuzzy number arithmetic operations between the trapezoidal fuzzy numbers 1 ~

A

and

2 ~

A

are as follows :

Fuzzy numbers addition

:

(

a

1,b1,c1,d1)

(

a

2,b2,c2,d2) = (

a

1+

a

2,b1+b2,c1+c2,d1+d2).

Fuzzy numbers subtraction

:

(

a

1,b1,c1,d1)

(

a

2,b2,c2,d2) = (

a

1-d2, b1-c2,c1-b2,d1-

a

2) . For example : Let 1

~

A

and 2 ~

A

be two trapezoidal fuzzy numbers , where 1 ~

A

= (16,20 ,22,24) and 2

~

A

= (3,4,5,6) . Then,

1 ~

A

2

~

A

= (16, 20, 22,24)

(3,4,5,6) = (19, 24, 27,30)

1 ~

A

2

~

A

= (16, 20, 22,24)

(3,4,5,6) = (10,15,18,21). If a trapezoidal fuzzy number

~

A

= (a,b,c,d) is symmetric (i.e., b=c) then the fuzzy number is called triangular fuzzy number.

2.2. Metric Distance ranking of fuzzy numbers

Chen and Cheng [41] proposed a metric distance method to rank fuzzy numbers (comparison of fuzzy numbers). Let ~

A

and ~

B

be two fuzzy numbers defined as follows :



~ ~ ~ ~ ~

,

)

(

,

)

(

)

(

A R A A L A

A

f

x

x

m

m

x

x

f

x

f

(1)



~ ~ ~ ~ ~

,

)

(

,

)

(

)

(

B R B B L B

B

f

x

x

m

m

x

x

f

x

f

(2)

where ~ A

m

and ~ B

m

are the mean of ~

A

and

~

B

. The metric distance between ~

A

and ~

(4)

2 1 1 0 2 1 0 2 ~ ~

)

(

)

(

)

(

)

(

)

,

(

~ ~ ~ ~

g

y

g

y

dy

g

y

g

y

dy

B

A

D

R B R A L B L A

, (3)

where L

A

g

~ ,

R A

g

~ ,

L B

g

~ and

R B

g

~ are the inverse functions of L A

f

~ ,

R A

f

~ ,

L B

f

~ and

R B

f

~ , respectively.

In order to rank fuzzy numbers, Chen and Cheng [41] let the fuzzy number

~

B

= 0 then the metric distance between

~

A

and 0 is calculated as follows : 2 1 1 0 2 1 0 2 ~

)

(

)

(

)

0

,

(

~ ~

g

y

dy

g

y

dy

A

D

R A L A (4)

The larger the value of D( ~

A

,0), the better the ranking of ~

A

.

According to [41], a trapezoidal fuzzy number ~

A

=

(

a

1

,

a

2

,

a

3

,

a

4

)

can be approximated as a symmetry fuzzy

number S[,], denotes the mean of ~

A

,  denotes the standard deviation of ~

A

, and the membership function of ~

A

is defined as follows :



x

if

x

x

if

x

x

f

A

,

)

(

,

)

(

)

(

~ (5)

where  and  are calculated as follows :

4

)

(

2

a

4

a

1

a

3

a

2

, (6)

4

4 3 2

1

a

a

a

a

. (7)

If

a

2

a

3 , then A becomes a triangular fuzzy number, where ~

A

=

(

a

1

,

a

2

,

a

4

)

and  and  can be calculated as follows :

,

2

1 4

a

a

(8)

4

2

2 4

1

a

a

a

(9)

The inverse functions L

A

g

~ and

R A

g

~ of

L A

f

~ and

R A

f

~ respectively, are shown as follows :

g

L

y

y

A

)

(

~ (10)

g

R

y

y

A

)

(

(5)

2.3 Fuzzy CPM Based on Metric Distance ranking of Fuzzy Numbers[12]

The operation time for each activity in the fuzzy project network is characterized as a positive trapezoidal fuzzy number. In accordance with CPM, the forward pass yields the fuzzy earliest-start and earliest-finish times :

j s j i P j s

i

E

t

E

~ ~ ) ( ~

max

(12)

i s i f

i

E

t

E

~ ~ ~

(13)

where

s i

E

~

is the fuzzy earliest –start time with

s A

E

~

= (0,0,0) at the initial node i = A ,

f i

E

~

is the fuzzy earliest finish time with

f Z

E

~

equal to the fuzzy project network completion time ~

T

at the ending node i = Z, P(i) is the set of predecessors for activity i , and

t

i

~

is the operation time for activity i.

The backward pass is performed to calculate the fuzzy latest-start and latest-finish times :

j f j i S j f

i

L

t

L

~ ~ ) ( ~

min

(14)

i f i s

i

L

t

L

~ ~ ~

(15)

where

f i

L

~

is the fuzzy latest-finish time with

f Z

L

~ = ~

T

at the end node i = Z ,

s i

L

~

is the fuzzy latest start time and S(i) is the set of successors for activity i . Once

s i

E

~ , f i

E

~ , s i

L

~ and f i

L

~

have been determined for the ith activity, the fuzzy float time is either

s i s i F

i

L

E

T

~ ~

(16) or f i f i F

i

L

E

T

~ ~

(17)

We can easily compute the fuzzy float times of all activities in a project network. In Crisp CPM, activity i is said to be a critical activity if its float time is zero. This concept implies that the criticality rises as the fuzzy float time decreases.

Fuzzy Critical Path Method [12] :

Consider the fuzzy project network, where the duration time of each activity in a fuzzy project network is represented by a trapezoidal fuzzy number.

.Step 1 : Calculate

s i

E

~ ’s and f i

E

~

’s using Eq. (12) and Eq. (13) Step 2 : Calculate

f i

L

~ ’s and s i

L

~

’susing Eq. (14) and Eq. (15)

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Step 5 : Rank the total slack fuzzy time of each path using metric distance ranking .

Step 6 : The path having minimum rank in step 5 is the critical path.

Example :

Fig.1 shows the network representation of a fuzzy project network. Table I represents the total float of each activity in the fuzzy project network.

Fig. 1 A fuzzy project network

Table I : Total float of each activity in the fuzzy project network

Activity Fuzzy Activity time Total float

1-2 (10,15,15,20) (-160,-60,60,160) 1-3 (30,40,40,50) (-130,-35,75,170) 2-3 (30,40,40,50) (-160,-60,60,160) 1-4 (15,20,25,30) (-110,-20,95,185) 2-5 (60,100,150,180) (-100,-10,100,190) 3-5 (60,100,150,180) (-160,-60,60,160) 4-5 (60,100,150,180) (-110,-20,95,185) The possible paths of fuzzy project network (Fig.1) are 1-2-3-5 , 1-2-5 , 1-3-5 and 1-4-5.

Metric distance rank of total fuzzy slack time for each path in fuzzy project network (Fig. 1) are computed and presented in table II.

1

3

2

5

(7)

Table II : Metric distance rank of total fuzzy slack time for each path in fuzzy project network

Path Total fuzzy slack time (a,b,c,d)

Metric distance rank

1-2-3-5 (-480,-180,180,480) 24.49

1-2-5 (-260,-70,160,350) 53.38

1-3-5 (-290, -95,135,330) 31.58

1-4-5 (-220,-40,190,370) 79.48

Here, the path having minimum rank is 1-2-3-5. Therefore, the required critical path for the fuzzy project is 1-2-3-5. 3. Proposed Fuzzy Critical Path Method

In this paper, to find fuzzy critical path method, WCR method is used. WCR has been described in [42] and was used by [43]. This rule answers not only which one of two fuzzy numbers is greater but also what is the degree to which one fuzzy number is greater than another. Assuming that

~

A

and ~

B

are triangular fuzzy numbers that define as

(

1

,

2

,

3

)

~

a

a

a

A

and

(

1

,

2

,

3

)

~

b

b

b

B

. The degree to which is ~

A

greater than ~

B

is denoted by R ( ~

A

~

B

) and calculated as follows :

(

2

)

(

2

)

4

1

)

(

1 2 3 1 2 3

~ ~

b

b

b

a

a

a

B

A

R

(18)

Using this degree, we can define three relations between ~

A

and ~

B

: ~

A

B

~ if R ( ~

A

B

~ )  0 (19) ~

A

> ~

B

if R ( ~

A

> ~

B

) > 0 (20) ~

A

~ ~

B

if R ( ~

A

B

~ ) = R (

~ B

~

A

) = 0 (21)

If equation (19) holds, we will say that ~

A

is weakly greater than or equal to ~

B

.

WCR’s formula to calculate R ( ~

A

B

~ ) is linear. Most of ranking fuzzy numbers methods such as Chen and Cheng method[41] have not this linearity feature.

Proposed Fuzzy critical path algorithm

Step 1 : Identify activities in a project

Step 2 : Establish precedence relationships of all activities.

Step 3 : Estimate the fuzzy activity time with respect to each activity time.

Step 4 : Construct the fuzzy project network with triangular fuzzy numbers as fuzzy activity times.

Step 5 : Find all possible paths (n) in fuzzy project network

Step 6 : Add all the fuzzy activity times in each path using addition of fuzzy numbers which gives fuzzy path length in fuzzy number.

Step 7 : Find maximum fuzzy path (M) using WCR among all paths obtained in step 6.

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itself.

Step 9 : Arrange the paths in ascending order(

1

,

2

,...,

n) using WCR ,

according to the values obtained .

Step 10 : Assign the total float of each activity by the following :

Choose the path 1 , assign the path value of 1 as the total float of each

activity in that path.

Choose path 2, assign the path value of 2 as the total float of each activity

in the path discarding the activities already assigned.

Continue the process until all the activities assigned the float time .

Step 11: Find the Fuzzy Total float of each path by adding the total float of each activity in the path.

Step 12: Rank the fuzzy total float of each path based on fuzzy ranking method.

The path with least rank is called critical path of the fuzzy project network. 4. An Example

Fig.2 shows the network representation of a fuzzy project network. Table III represents the activity times represented by triangular fuzzy numbers in the fuzzy project network presented in Fig 2.

Fig. 2 Fuzzy Project network

1

2

3

4

5

6

7

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Table III Activities and their Fuzzy durations

The calculations of step 5 to step 9 of the proposed fuzzy critical path method are presented in Table IV.

Table-IV Computation results of path ranking

Path Total fuzzy activity time of the path

(a,b,c)

M

Ө (a,b,c) Path Rank 1-2-4-6-8 (8,15,21)

M

(-13,0,13) P1

1-2-4-7-8 (9,14,21) (-13,1,12) P2

1-3-5-7-8 (6,13,19) (-11,2,15) P3

1-3-6-8 (5,13,17) (-9,2,16) P4

Fuzzy total float time of each activity for fuzzy project network is presented in table V.

Table-V : Computation results of Fuzzy Total float of each activity

The fuzzy critical path 1-2-4-6-8 is obtained using steps 11 and 12 of the proposed fuzzy critical path method. Conclusion

A new approach for finding total float of each activity, critical activities and critical path in a fuzzy project network has been proposed. The method proposed in this paper is more effective and easy to determine the activity criticalities, finding the critical path to compare with other fuzzy critical path methods.

References

[1] F.S. Hiller, G.J. Lieberman, Introduction to operations research , Seventh edition, Mc Graw-Hill, Singapore, 2001.

[2] L.J. Krajewski, L.P. Ritzman, Operations Management : process and value chains, seventh edition, Prentice-Hill, New Jersey, 2005.

Activity Fuzzy activity time 1-2 (2,3,4) 1-3 (1,3,4) 2-4 (1,3,5) 3-5 (1,2,3) 3-6 (2,5,7) 4-6 (3,4,6) 4-7 (3,4,5) 5-7 (1,4,5) 6-8 (2,5,6) 7-8 (3,4,7)

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1-2 (2,3,4) (-13,0,13)

1-3 (1,3,4) (-11,2,15)

2-4 (1,3,5) (-13,0,13)

3-5 (1,2,3) (-11,2,15)

3-6 (2,5,7) (-9,2,16)

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4-7 (3,4,5) (-13,1,12)

5-7 (1,4,5) (-11,2,15)

6-8 (2,5,6) (-13,0,13)

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[25] Mon D L, Cheng CH, Lu H C. Application of fuzzy distributions on project management. Fuzzy Sets and Systems 1995,73, 227-234. [26] Yao J S, Lin F T. fuzzy critical path method based on signed distance ranking of fuzzy numbers. IEEE Transactions on Systems, Man, and

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[29] Yakhchli S H, Ghodsypour S H. Erratum to “On computing the latest starting times and floats of activities in a network with imprecise durations”. Fuzzy Sets and Systems 2008 a ; 159:; 856.

[30] Dubois D, Fargier H, Fortin J. Computational methods for determining the latest starting times and floats of tasks in interval-valued networks. Journal of Intelligent manufacturing 2005; 16;407-421.

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The capacities of the arcs and the fuzzy costs (modal value and spreads) of the commodities are given in Table 11, and the solution of the fuzzy problem found by decomposition

Fuzzy Newton-Cotes formula, such as fuzzy trapezoidal method and fuzzy Simpson method are calculated by integration of fuzzy functions on two and three equally space points.. Also

The questions discussed in inverse DEA can be considered with fuzzy data, that is, assume that some data are fuzzy numbers and we increase some or all input levels of a given DMU

To improve this and to facilitate supplier selection process, the paper discusses a fuzzy - AHP approach using triangular fuzzy numbers to represent decision makers’ comparison

RISK ASSESSMENT IN PROJECT PLANNING USING FMEA AND CRITICAL PATH METHOD.. Sandra Milena

In this paper we discuss fuzzy strongly continuous semigroups fuzzy dynamical systems relationships and results obtained in [1] on invariant sets and stability of such fuzzy sets,

It was also showed that project network its fuzzy earliest-start time, fuzzy earliest-finish time, fuzzy latest-start time and latest-finish time of each activity in Table