Application of fuzzy distributions on project management

FUZZY

sets and systems ELSEVIER

Fuzzy Sets and Systems 73 (1995) 227-234

Application of fuzzy distributions on project management Don-Lin Mon*, Ching-Hsue Cheng, Han-Chung Lu Department of System Engineering, Chung Cheng Institute of Technology, Tahsi 33509, Taiwan, ROC Received May 1994; revised July 1994

Abstract PERT/cost which is an extension of PERT to include economic considerations brings cost factors into project control decisions. However, PERT/cost is usually developed by project managers without enough experience. In calculating the critical path of crisp PERT/cost network, we usually suppose that all activity durations follow beta distribution of three time parameters (ai,m,,b~), i.e., the mean and variance of an activity duration are (a t + 4m i + b~)/6, ( b ~ - ai)2/36, respectively. Basically, the activity durations and their distributions are subjectively determined, so the beta distribution is not always applicable. Therefore, we propose a fuzzy PERT/cost that can be applied to a variety of fuzzy distributions of activity durations. Intuitively, the higher the risk level, the more uncertainty in time/cost is involved in the project. From fuzzy set theory, it corresponds to lower confident level (or-cut) or larger interval value. We will use such interval of confidence to represent fuzzy activity durations so that ~t-cut can be interpreted as a risk level and the attitude of the decision makers can be expressed by another index of optimism, 2. Thus, the relationships between project time and cost under different risk levels and different degrees of optimism can be obtained. The solution procedure will be described in detail in conjunction with an example to illustrate the analysis, algorithm and computation of the proposed method. We have verified two of the empirical philosophy that for the higher risk level, the shorter project time for the optimist (2 = 1) could be obtained; on the other hand, for the lower risk level, the shorter project time was obtained for the pessimist (2 = 0). We have observed that no matter whatever the risk level is, the results of the project time and cost of fuzzy PERT/cost at 2 = 0.5 (neither optimistic nor pessimistic) are equal to those of the crisp PERT/cost. Keywords: PERT/cost; Fuzzy PERT/cost; Fuzzy distribution; Risk level; Degree of optimism

1. Introduction P E R T / c o s t , an extension of P E R T to include e c o n o m i c considerations, takes cost factors into project control considerations. A cost network can be superimposed u p o n a P E R T network to estimate the total cost and the cost slope for each

* Corresponding author.

activity. P E R T offers a way of dealing with r a n d o m variation and allows for change in the scheduling of activities. While P E R T / c o s t enables the decision m a k e r to evaluate alternatives relative to the allocation of resources for activity accomplishment. In m a n y instances, time can be saved by applying m o r e resources or, conversely, cost m a y be reduced by extending the time required to complete an activity. P E R T / c o s t has the following characteristics.

0165-0114/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved SSDI 0 1 6 5 - 0 1 1 4 ( 9 4 ) 0 0 3 0 9 - 2

D.-L. M o n e t al. / Fuzzy Sets and Systems 73 (1995) 227 234

228

(1) PERT/cost network is used to portray the interrelationships among the activities associated with a project. Each activity duration is an independent random variable in the PERT/cost network, with its own probability density function and statistical characteristic. (2) Time value of PERT/cost is defined as a random variable with a beta distribution and ranged from ai to bi with mode m~, for each activity. The expected time and the variance of the activity duration are defined as, respectively, ti =

ai d- 4 m i q- bi

6

,

(b i - ai) 2

Var(ti) = a] - - - , 36

where ai is the optimistic time, mi the most probable time, and b~ the pessimistic time. (3) It is assumed that the critical path was based on the activities performed in normal time with a normal allocation of resources. This is called a normal schedule. We can obtain the normal estimate of time and the normal estimate of cost. When one or more activities are performed with additional resources to shorten their time durations, a crash schedule is said to exist. We can obtain the estimated crash time and estimated crash cost. The times and costs required to complete the various activities in a project are generally not known a priori. Thus, PERT/cost incorporate uncertainties in time and cost in the analysis. In real applications there is a great need to handle situations on a higher uncertainty level and the structure of the three parameters as mentioned above can be imprecise wherein the fuzzy set theory is well suitably applied. Hence, fuzzy PERT/cost approach seemingly becomes a more appropriate way to illustrate and realize time and cost estimation. The Fuzzy PERT (FPERT) was originated from Chanas and Kamburowski [2]. They presented the project completion time in the form of fuzzy sets in the time space. Based on the given possibility distributions of activity durations, the possibility distribution of the project completion time can be derived. From [1], initiated another F P E R T method which, however, neither contains the definitions of the required possibility distribution nor gives a detailed algorithm for finding the distribution for project duration. Gazdik [3] developed

a fuzzy network of an a priori unknown project to estimate the activity durations, and used fuzzy algebraic operators to calculate the duration of the project and its critical path. The initial uncertainty about the project input parameters was reduced to quasi-deterministic output data. Recently, Wong [13] applied the vertex method [10] to compute FPERT, the interval analysis in forward sweeps through the project network was essentially the same as that derived by the extension principle method. The execution of the FPERT equations should lead to results with bounds which are larger than the corrects bounds, due to the multiple occurrences of the independent variables. Different distributions [5,6,9] have been proposed, for example, the probability distributions for activities durations may take triangular distribution, Poisson distribution, or normal distribution. Observing no general possibility distribution for activity duration in the project and no cost parameter considered in FPERT researches yet, we propose this paper on fuzzy PERT/cost with different fuzzy distributions on activity duration [8] under various a-level and index of optimism. Intuitively, the higher the risk level, the more uncertainty in time/cost is involved in the project. From fuzzy set theory, it corresponds to lower confidence level (a-cut) or larger interval value. So we considered a-cut as risk level and set [14] ~ < 0.3, as high risk; 0.3 ~< ct < 0.7, as medium risk; ct ~> 0.7, as low risk. We will use the interval arithmetic to represent the fuzzy activity durations and the a-cut as a risk level and, 2, an index of optimism as the attitude of the decision makers, a larger 2 indicates a higher degree of optimism (less time required), vice versa. This way we create a more practical and flexible analysis and planning appropriated for the real world project management.

2. Preliminaries

In this section we present some definitions related to the fuzzy set theory [4] and the index of optimism 2. Definition 2.1. A real fuzzy number ,4 is a fuzzy subset of the real line R with membership function

D.-L. Mon et al. / Fuzzy Sets and Systems 73 (1995) 227-234

/~i as defined in the following: (a) /~a is a continuous function from II~ on a closed interval [0, 1], (b) #,i = 0 for all x ~ ( - ~ , a], (c) #a is strictly increasing on [a, hi, (d) /~,i = 1 for all [b, c], (e) ~i is strictly decreasing on [c, d], (f) # ~ = 0 for all x ~ [ d , ~ ) , where - ~
I~;dx) =

I'L

wla~(x),

a <~ x <~ b,

w,

b <~ x <~ c, O < w <~ l,

g

wl~;4(x)

c <<.x ~ d,

0,

otherwise,

L.

R

where It~i.[a,b] --* [0,1], and /~a "[c,d] ~ [ 0 , 1 ] . When w = 1, it is called the normal fuzzy number. Definition 2.2. Fuzzy subset A c It~ is convex iff every ordinary

subset is convex; that is, if it is a closed interval of ~. Definition 2.3. Suppose that A is fuzzy number with membership function /~j, then for every c~e [0, 1], the set A~ = { x I #,i(x) >~ ct}, is called an c~-cut of/~. Definition 2.4. The arithmetic of fuzzy numbers depends on the arithmetic of the interval of confidence at level ~. Some main operations for positive fuzzy numbers described by the interval of confidence are VaL,aR,bL, bRE ~,

A~ = [a~.,a[],

/~ = [b[, b~], ct e [0, 1], A( + )B = [a~ + b[,a~ + b~], A(-)B

= [a[ - b~,a~ - b[],

A(" )B = [a[ b~, a~tb~], A ( + ) B = [ a [ / b [ , a l i b i ].

229

Definition 2.5. If A is a fuzzy number with membership function /~a, and A~ = [a[,a~], a~ = 2a~. + (1 -- 2)a] can represent an element in the universe of discourse determined by an index of optimism 2. The index of optimism 2 indicates the degree of optimism of a decision maker. A larger 2 indicates a higher degree of optimism.

3. PERT/cost for activity durations with different fuzzy distributions Because of the uncertainty involved in the estimation of the activity durations, we assume two simple fuzzy distributions of durations in this fuzzy PERT/cost analysis. We will estimate the activity time and the crash time by either fuzzy triangular distributions (FTD) or fuzzy normal distributions (FND). However, the normal cost and the daily incremental crash cost will use crisp values. Before the proposed method is introduced, some of the notations are defined as in the following. Notations aij the activity from node i to node j t normal duration of activity ai~ t' crashing time of activity a~ Te the earliest occurrence time of event, where TE = max {Er,,Er . . . . . . EF,} TL the last permission occurrence time of event, where TL = m i n { L s ~ , L s . . . . . . Ls.} Es the earliest starting time of activity, where Es = TE EF the earliest finishing time of activity, where Ev = Es + t Ls the latest starting time of activity, where Ls = Lv - t LF the latest finishing time of activity, LF = TL tc the crashing time of activity air, where tc = t -- t' C,, the crashing costs of activity a~j C. the crashing costs of activity a~j for per unit time Algorithm From above definitions and notations, if the corresponding risk level (s-cut) is fixed, then the critical path of the fuzzy PERT/cost's network is dependent upon the index of optimism 2.

230

D.-L. &Ion et aL / Fuzzy Sets and Systems 73 (1995) 227-234

The computational procedures are summarized as follows. [Step 1] Choose a risk level 0t and compute the a-cuts of all the fuzzy distribution inputs to the FPERT/cost problem. [Step 2] Estimate the degree of satisfaction of the decision maker by using the index of optimism, 2, to determine the three time parameters. [Step 3] Calculate the completion time and the critical path using the three time parameters of the project. (A) Forward calculate the times from starting node to ending node, the principles are (i) TE = 0 (the earliest occurrence time of starting node) (ii) Es = TE (the precursory event), g t = Es + t (iii) Te = max{Er,,Ee . . . . . . Ee.}. If an event has n activities converging to it, then return to (ii). (B) Backward calculate the times from ending node to starting node, the principles are (iv) TL = Te (ending node). (v) L e = Tt. (the successive event), L s = L r - t, (vi) TL = min {Ls,, L s . . . . . . Ls.}. If an event with n activities diverging from it, then return to

(v). (C) Calculate the floating time for all activity that is L s - L r . (D) To identify project's critical path by tracing activities which have zero floating time in sequence of the events. [Step 4] Calculate two types of floats, i.e., total float (Tf) and the free float (Ff) by formulae

alj,

the the the the

Tf --- Le -- EF, F f -~ Es

(of the earliest starting time of the very next activity) - Ee (of the earliest finishing time of the activity).

Remark 1-13]. Float is a measure of allowable delay or leeway. Total float is the time that a project can be delayed before it affects the completion time. Free float is the time that a task may be delayed before it affects the earliest starting time of any of the activities immediately followed. A critical path of the PERT can be identified either by finding the

longest path using activity durations directly or by the fact that their total floats equal to zero. [Step 5] Implementing fuzzy PERT/cost by the time-cost option. The incremental cost (I~) is an important index of this time--cost option which is too cumbersome to be described here [11]: Ic =

crash c o s t - normal cost normal t i m e - crash time"

4. Numerical example The fuzzy PERT/cost network is configured with six events and seven activities in this example [11] as shown in Fig. 1. The assumed distributions of FTD and FND of the various activities are tabulated in Table 1. As we assume the risk level to be 0.05, the time distributions and crash time distributions can be reduced to Table 2 by the use of the interval of confidence. Tables 3 and 4 can be obtained in the same way for ~ = 0.5 and 0.8, respectively. As the index of optimism 2 = 1, 0.5, and 0 is introduced in the computation, the relative results of critical paths and project times are summarized as in Table 5. The final results of time-cost options with respect to different ~ are shown in Tables 6-8. We can find the relationship between project time and project cost for different risk levels under different degree of optimism as in Figs. 2-4. From Table 5, we verified two of the empirical philosophy that for the higher risk level, the shorter project time for the optimist (;~ = 1) could be obtained; on the other hand, for the lower risk level, the shorter project time for the pessimist (2 = 0) was obtained. We also obtained the same critical path (A,B,D,F,G) when 2 = 0.5 or higher, no

SO~;lOOO~ Fig. 1. Fuzzy PERT/cost network.

231

D.-L. Monet al. / Fuzzy Sets and Systems 73 (1995) 227-234

Table 1 Fuzzy time distribution and crash time distribution plus daily incremental crash cost Activity

Fuzzy activity time distribution

Fuzzy crash time distribution

Normal cost

Daily incremental crash cost

0-1 (A)

1 day (crisp time)

1 day (crisp time)

$5000

Not applicable

12(B)

/ ~ ( x ) = e (x-3)2, x > 0

#a(x)=e -(x 2 ) : , x > 0

1-3 (C)

/~e(x) = I e(x 7), x ~< 7 e (x 7), x > 7

l~e3(x-4), x ~<4 /re(x)= l e-3(:'-4), x > 4

2-3(0)

[½(x-2), /~t~(x)=ll, ½(8

L

2~
ff½(x-1),

l~
4~
/a~(x)=[½(5_x),

3~
2-4 (El

#~(x) = e -(1/3)(x-8)~, x > 0

/a~(x) = e (x 6)1, x > 0

3-4 (F)

)'½(x-2), 2 ~ < x < 4 #p(x) = [ ½ ( 6 - x), 4 ~< x <~ 6

#p(x) =

4-5 (G)

/~d(x) = [ e (4/31(x 1)

~e (4/3)(x-1),

x ~< 1 x> 1

{~-1, x,

l~
~e3(X 1),

5000

$7000

11000

2000

10000

1000

8500

2000

8500

4000

x ~< 1

#d(x)= [ e -3(x-l)

x> 1

Not applicable

5000

Table 2 The activity times and crash times when ~ = 0.05

Table 4 The activity times and crash times when ~ = 0.8

Activity

Activity time

Crash time

Activity

Activity time

Crash time

o-1 (A)

[1, 1]

[1, 1]

1-2 (B) 1-3 (C) 2 3 (D) 2-4 (E) 3-4 (F) 4-5 (G)

[1.2692,4.7308] [4.0043,9.9957] [2.1,7.9] [7.0007, 8.9993] [2.1,5.9] [0.2511,1.7489]

[0.2692,3.7308] [3.0014,4.9986] [1.1,4.9] [4.2692, 7.7308] [1.05,2.95] [0.0014,1.9986]

0-1 1-2 1-3 2-3 2-4 3-4 4-5

[1, 1] [2.53,3.47] [6.78, 7.22] [3.6, 6.4] [7.18,8.82] [3,6,4.4] [0.83, 1.17]

[1,1] [1.53,2.47] [3.93, 4.07] [2.6, 3.4] [5.53,6.47] [1.8,2.2] [0.93, 1.07]

(A) (B) (C) (D) (E) (F) (G)

Table 5 The summary of the relative results with different ~ and ,:o Table 3 The activity times and crash times when a = 0.5 Activity

Activity time

Crash time

0 1 (A) 1-2 (B) 1 3 (C) 2-3 (D) 2-4 (E) 3-4 (F) 4-5 (G)

[1,1] [2.17,3.83] [6.31,7.69] [3,7] [6.56,9.44] [3,5] [0.48,1.52]

[1,1] [1.17,2.83] [3.77,4.23] [2,4] [5.17,6.83] [1.5,2.5] [0.77,1.23]

~t

2

Critical path

Project time

0.05 0.05 0.05 0.5 0.5 0.5 0.8 0.8 0.8

1 0.5 0 1 0.5 0 1 0.5 0

A, B, E, G A, B, D, F, G A, B, D, F, G A, C, F, G A, B, D, F, G A, B, D, F, G A, C, F, G A, B, D, F, G A, B, D, F, G

9.52 14 21.28 10.79 14 18.35 12.21 14 16.44

D.-L. Monet al. / Fuzzy Sets and Systems 73 (1995) 227-234

232

Table The

6 time-cost

options

with

~ = 0.05

2=0

Project

time

2=0.5

Additional

Project

cost

2=

Additional

cost (least)

Project

cost

Additional

cost (least)

--

53000

18

3000

56000

--

--

17

8000

64 000

.

.

.

.

16

6000

70 000

.

.

.

.

15

9000

--

--

--

--

53000

--

--

--

--

13

--

--

1000

54 000

12

--

--

5000

59 000

--

6000

65000

--

74 000

--

11 I0

--

--

9 (9.52)

9000

.

.

.

--

.

-53 000

8

.

.

.

.

2000

55 000

7

.

.

.

.

6000

61 000

Table The

7 time-cost

options

with

• = 0.5

2=0

Project

time

2 =0.5

Additional

Project

cost

2 = 1

Additional

cost (least)

Project

cost

Additional

cost (least)

--

53000

16

3000

56000

15

6000

62 000

.

.

14

6000

68 000

.

.

13

9000

77 000

14

--

--

--

53000

13

--

--

1000

54000

--

12

--

--

5000

59 000

--

11

--

--

6000

65 000

--

10

--

--

9000

74000

--

--

11 ( 1 0 . 7 9 )

.

9 8

--

.

.

.

.

.

--

.

.

matter what the risk level is. From Fig. 3, we observed that the results of the project time and cost of the fuzzy PERT/cost at 2 = 0.5 are independent of the levels of risk and are equal to those of the crisp PERT/cost [11].

--

--

---

---53000

-.

--

. --

-.

cost

---

.

Project

cost (least)

19 (18.35)

10

cost

--

79 000 --

Project

cost (least)

21 (21.28)

14

1

2000

55000

6000

61 000

9000

70000

5. Conclusion Considering the problem of project management in the real world, this research is devoted on the fuzzified PERT/cost analysis of activity durations

233

D.-L. Monet al. / Fuzzy Sets and Systems 73 (1995) 227 234

Table 8 The time-cost options with c~= 0.8 2=0 Project time

2=0.5

Additional cost (least)

17 (16.44) 15 14 13 12

Project cost

Additional cost (least)

53000 55 000 62 000 68 000 74 000

2000 7000 6000 6000

14

~.= 1 Project cost

--

---

--

53 000 54000 59000 65 000 74 000

1000 5000 6000 9000 . --

-.

.

--

.

.

.

.

-~ 53000 55000 61 000 70 000

. 2000 6000 9000

--

--

.

Project cost

---

--

13 12 11 10 13 ( 1 2 . 2 1 1 12 11 10

Additional cost (least)

.

80

100

75 80 7O

I

o..

•

20

a.

60

• 0t = 0.05 I

0t = 0 . 0

ot = 0.5

ea

•a=0.8

c~ =0.8 0

-~

~

I.

_x

,

A

,

,

,

,

50

I

14 21.3

19 18.4 17 16.4

16

15

14

13

12

Project Time Fig. 2. The relationship between project time and cost for different ~ under 2 = 0. with different fuzzy d i s t r i b u t i o n s . I n o r d e r to o b t a i n the critical p a t h of the fuzzy P E R T / c o s t , we i n t r o d u c e the i n t e r v a l of c o n f i d e n c e to r e p r e s e n t the a c t i v i t y d u r a t i o n s a n d i n t e r p r e t e d the s - c u t as a risk level a n d e x p r e s s e d the a t t i t u d e of the decis i o n m a k e r b y a n i n d e x of o p t i m i s m 2. W e des c r i b e d the s o l u t i o n p r o c e d u r e s in d e t a i l a n d used a n e x a m p l e to i l l u s t r a t e the a n a l y s i s , a l g o r i t h m a n d c o m p u t a t i o n of the p r o p o s e d m e t h o d . W e o b t a i n e d

I

0.5 }

55

I 13

I

I

12

11

10

Project Time

Fig. 3. The relationship between project time and cost for different ~ under 2 = 0.5.

the r e l a t i o n s h i p b e t w e e n the p r o j e c t t i m e a n d cost at different risk levels a n d u n d e r v a r i o u s degree of o p t i m i s m . W e o b s e r v e d t h a t n o m a t t e r w h a t the risk level is, the results of the p r o j e c t t i m e a n d cost of fuzzy P E R T / c o s t at 2 = 0.5 are e q u a l to t h o s e of the crisp P E R T / c o s t . W e also verified t w o of the e m p i r i c a l p h i l o s o p h y t h a t the h i g h e r the risk level, the p r o j e c t t i m e for the o p t i m i s t (2 = 1) b e c a m e shorter, a n d vice versa, for the l o w e r risk level, the

234

D.-L. Monet al. / Fuzzy Sets and Systems 73 (1995) 227-234 75

70

•

55

•

•

• I

50

12.21 12

I

11

I

I

• ~--0.5 I

• I

10 10./9 10

Orol~t T~'~

I

I

I

I

9

9.52

8

7

• (x=0.8

]

Fig. 4. The relationship between project time and cost for different ct under 2 = 1.

project time became shorter for the pessimist (2 = 0). Because the optimist preferred higher risk level to lower risk who can invest the most of the resources to complete the project as soon as possible and to gain the maximum profit. On the contrary, the pessimist is more conservative than the optimist, who prefer the lower risk level to the higher level.

References I-1] J.J. Buckley, Fuzzy PERT, in: G.W. Evans et al., Eds., Applications of Fuzzy Set Methodologies in Industrial Engineering (Elsevier, Amsterdam, 1989) 103-114.

[2] S. Chanas and J. Kamburowski, The use of fuzzy variables in PERT, Fuzzy Sets and Systems 5 (1981) 11-19. [3] I. Gazdik, Fuzzy-network planning-FNET, IEEE Trans. Reliability R-32 3 (1983) 304-313. [4] A. Kaufmann and M.M. Gupta, Introduction to Fuzzy Arithmetic Theory and Application (Van Nostrand Reinhold, New York, 1991). [5] T.C.T. Kotiah and N.D. Wallace, Another look at the PERT assumptions, Management Sci. 20 (1973) 44-49. [6-1 K.R. MacCrimmon and C.A. Rayvec, An analytical study of the PERT assumptions, Oper. Res. 12(1) (1964) 16-37. [7] D.L. Mon and C.H. Cheng, Fuzzy system reliability analysis for components with different membership functions, Fuzzy Sets and Systems 62 (1994) 145-157. [8-1 D.L. Mon and C.H. Cheng, Evaluating weapon system by analytical hierarchy process based on fuzzy scales, Fuzzy Sets and Systems 62 (1994) 1-10. [9] W.H. Parks and K.D. Ramsing, The use of the compound poisson in PERT, Management Sci. 15 (1969) B397-B402. [10] H. Prade, Using fuzzy set theory in a scheduling problem: a case study, Fuzzy Sets and Systems 2 (1979) 153-165. [11] R.J. Thierauf and R.C. Klekamp, Decision Making Through Operations Research (Wiley, New York, 2nd ed., 1975). [12] G.E. Whitehouse, Systems Analysis and Design Using Network Techniques (Prentice-Hall, Englewood Cliffs, N J, 1973). [13] F. Wong, Modeling and analysis of uncertainties - a personal view, Proc. Fuzzy Set in Engineering Applications, Department of Civil Engineering, National Central University, Republic of China (1993). [14] C.T. Yang, Weapon system development and risk management, Proc. 2nd National Conf. on Science and Technology of National Defense, Chung Cheng Institute of Technology, Republic of China (1993).