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Applying fuzzy method for measuring criticality in project network
Information Sciences 177 (2007) 2448–2458 www.elsevier.com/locate/ins
Applying fuzzy method for measuring criticality in project network Chen-Tung Chen a
a,*
, Sue-Fen Huang
b
Department of Information Management, National United University 1, Lien Da, Kung-Ching Li, Miao-Li 36003, Taiwan b Department of Information Management, Da-Yeh University 112, Shan-Jiau Road, Da-Tsuen, Changhua, Taiwan Received 24 January 2007; accepted 25 January 2007
Abstract Program evaluation and review technique (PERT) is widely used as a tool for managing large-scale projects. In the traditional PERT model, the durations of activities (tasks) are either represented as crisp numbers or drawn from the beta distribution to estimate the task durations such as pessimistic, most likely and optimistic times. However, the operation time for each activity is usually difficult to define and estimate precisely in a real situation. The aim of this paper is to present an analytical method for measuring the criticality in a project network with fuzzy activity times. Triangular fuzzy numbers are used to express the operation times for all activities in a project network. A new model that combines fuzzy set theory with the PERT technique is proposed to determine the critical degrees of activities (tasks) and paths. In the proposed model, a possibility index is defined to identify the likelihood of meeting a specified required time for a project network. At the end of the paper, an example is presented to compare with those obtained using the proposed method as well as other methods. The comparisons reveal that the method proposed in this paper is more effective in determining the activity criticalities and finding the critical path. Ó 2007 Published by Elsevier Inc. Keywords: Project management; Fuzzy set; Program evaluation and review technique (PERT); Fuzzy PERT
1. Introduction In recent years, the range of project management applications has greatly expanded. Project management concerns the scheduling and control of activities (tasks) in such a way that the project can be completed in as little time as possible [1,29]. To ensure the project’s success, the project management team must identify the stakeholders, determine their needs and expectations, and manage those needs and expectations. A project network is defined as a set of activities that must be performed according to precedence constraints stating which activities must start after the completion of specified other activities [12]. Such a project network can be represented as a directed graph. There are two slightly different conventions for representing these network *
Corresponding author. Tel.: +886 37 381833; fax: +886 37 330776. E-mail address: [email protected] (C.-T. Chen).
0020-0255/$ - see front matter Ó 2007 Published by Elsevier Inc. doi:10.1016/j.ins.2007.01.035
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graphs, the activity-on-arrow (AOA) graph and the activity-on-node (AON) graph [30]. These conventions are illustrated in Figs. 1 and 2. In the AON graph, the nodes represent activities and the arcs represent precedence relations. A path through a project network is one of the routes from the starting node to the ending node. The length of a path is the sum of the durations of the activities on the path. The project duration equals the length of the longest path through the project network. The longest path is called the critical path in the network. A project is deemed complete if work along all paths is complete. After the durations and precedence relations of activities have been determined, project management techniques are used to calculate the completion time of the project. The most commonly used project management techniques are the Grant Chart, Milestone, Critical Path Management (CPM), and Project Evaluation and Review Technique (PERT) [8,27,31]. PERT is the most widely used management technique for planning and coordinating large-scale projects [2,6,11,12,14,16– 19,21,31]. It was developed in the 1950s to help managers schedule, monitor, and control large and complex projects [20]. Nowadays, it is extensively used both in industries and service organizations. By using PERT, managers are able to obtain [3,30]: 1. 2. 3. 4.
A graphical display of project activities (tasks). An estimate of how long the project will take. An indication of which activities are the most critical for timely project completion. An indication of how long any task can be delayed without delaying the project.
In traditional PERT techniques, various dynamic activity durations must be represented either as crisp numbers or as random variables drawn from certain probability distributions (normally, a Beta distribution). In PERT, we must specify three time estimates for each activity: an optimistic time, a most probable (or most likely) time, and a pessimistic time. The optimistic time is the fastest possible activity completion time for the given activity. The probable time is the time that will most likely be required. The pessimistic time is the ‘‘worst’’ task time that could be expected. The three time estimates are used to calculate both the expected completion time and variance for each activity [3,9,10,28]. The main drawback of PERT technique is the difficulty of obtaining the time estimates. It is hard to get precise information about activity durations in some situations, such as early rough planning in long range projects [21]. In a real world situation, operation times of activities in a project network may be difficult to define and estimate exactly. Therefore, it is important to compute the variance of the project completion time in a network [2,3]. In recent years many researchers have combined fuzzy-set theory with PERT to handle time estimates in project planning and control problems. The result is a new approach called fuzzy PERT (FPERT) [6,12– 15,18,19,24–26]. FPERT was first presented by Chanas and Kamburowski [4], who used fuzzy numbers to
1
a 3
4 c
2
b
Fig. 1. Activity on arc.
a c b
Fig. 2. Activity on node.
d
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represent activity durations in project networks. The possibility distribution of the project completion time is derived from the individual activity duration possibility distributions once they are known. All fuzzy activity durations are calculated based on the three time estimates. The fuzzy a-cut is then applied to calculate the upper and lower bounds on the completion time of the project network. One problem in Chanas and Kamburowski is that different values of a result in different bounds on the completion time. In this situation, there is no effective way to indicate the critical activities and paths for a project network. Mon et al. [26] remedied this problem by assuming that the duration of each activity is a e a ¼ ½aa ; aa , where positive fuzzy number. Using the a-cut of each fuzzy duration, they denoted the interval A L R a a aL and aR were the lower and upper duration bounds. Using traditional PERT, a linear combination of the duration bounds would be used to represent the operation time of each activity and determine the critical activities and paths. However, the a values would yield different critical activities and paths in this approach. Chanas and Zielinski [5] assume that the operation time of each activity can be represented as a crisp value, interval or a fuzzy number. The calculation procedure of Mon et al. [26] is similar. The method proposed in Dubois et al. [12] assigns a different level of importance to each activity on a critical path for a randomly chosen set of activities. All of these FPERT methods reduce uncertainties by using fuzzy numbers to represent the activity operation times. However, the procedures for calculating the critical degree of each activity and path are quite complicated. Moreover, no effective and straightforward method is available to estimate the possibility of meeting a specified requirement time. In fact, uncertainty is an attribute of information [33]. The subjective judgment or prediction of an expert for the times of activities may be ill defined. It is reasonable to represent them as linguistic variables or fuzzy numbers [33]. In this paper, a FPERT method is proposed to deal with completion time management and the critical degrees of all activities for a project network. First, the fuzzy completion time of a project network was derived from the fuzzy activity operation times. Second, an index to compute the critical degrees of activities and paths was employed. Then, a straightforward approach to identify the critical activities and critical paths in a project network was used. Finally, an index that determines the possibility of a project meeting a specified requirement time was developed. 2. Fuzzy sets and notations A fuzzy set can be mathematically constructed by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set [32,33]. This grade corresponds to the individual’s similarity to the concept represented by the fuzzy set. Thus, individuals may belong in the fuzzy set to a greater or lesser degree as indicated by a larger or smaller membership grade. As already mentioned, these membership grades are often represented by real numbers ranging from a minimum of 0 to a maximum of 1 [23,33]. e is a fuzzy set whose membership function l ðxÞ satisfies the following conditions [23]: The fuzzy number A e A
1. 2. 3.
leðxÞ is piecewise continuous; A leðxÞ is a convex fuzzy subset; A leðxÞ is the normality of a fuzzy subset, implying that for at least one element x0 the membership grade A must be 1, i.e. leðx0 Þ ¼ 1. A
e is defined as The a-cut of a fuzzy number A e a ¼ fxi : l ðxi Þ P a; xi 2 X g A eA
ð1Þ
where a 2 ½0; 1. e a ¼ ½aa ; aa , where aa e a represents a non-empty bounded interval contained in X, denoted A The symbol A L R L and aaR are the lower and upper bounds of the closed interval [22,34]. ~ and ~n are m ~ a ¼ ½mal ; mau and ~na ¼ ½nal ; nau (a 2 ½0; 1Þ. Based The a-cut of any two positive fuzzy numbers m on the interval of confidence [22], the following major operations can be applied to two positive fuzzy numbers ~ and ~ m n:
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µ~ T (x )
1
0
l
m
u
x
Fig. 3. Positive triangular fuzzy number Te .
a
~ ðmðþÞ~ nÞ ¼ ½mal þ nal ; mau þ nau ~ ðmðÞ~ nÞa ¼ ½mal nau ; mau nal
ð2Þ ð3Þ
Particularly common in practical applications are triangular fuzzy numbers, denoted Te ¼ ðl; m; uÞ. When l > 0, Te is a positive triangular fuzzy number (PTFN) [11,34]. Fig. 3 shows the membership function of a positive triangular fuzzy number Te , defined as 8 xl > < ml ; l 6 x 6 m ux le ðxÞ ¼ um ð4Þ ; m6x6u T > : 0; otherwise where l > 0. Given two PTFNs Te 1 ¼ ðl1 ; m1 ; u1 Þ and Te 2 ¼ ðl2 ; m2 ; u2 Þ, fuzzy addition and subtraction can be performed as follows [22]: Te 1 Te 2 ¼ ðl1 þ l2 ; m1 þ m2 ; u1 þ u2 Þ Te 1 H Te 2 ¼ ðl1 u2 ; m1 m2 ; u1 l2 Þ
ð5Þ ð6Þ
Many ranking methods have been developed to transform fuzzy numbers into crisp values [7]. Lee and Li [25] presented the generalized mean value method, an effective method for ranking and comparing fuzzy numbers. For a triangular fuzzy number Te ¼ ðl; m; uÞ, the generalized mean value Gð Te Þ and deviation Sð Te Þ are given by [25] Gð Te Þ ¼
lþmþu 3
ð7Þ
and 1 2 ½l þ m2 þ u2 lm lu mu 18 Two triangular fuzzy numbers Te 1 ¼ ðl1 ; m1 ; u1 Þ and Te 2 ¼ ðl2 ; m2 ; u2 Þ can be compared as follows: Sð Te Þ ¼
(1) If GðT 1 Þ > GðT 2 Þ then Te 1 > Te 2 . (2) If GðT 1 Þ ¼ GðT 2 Þ and Sð Te 1 Þ < Sð Te 2 Þ then Te 1 > (3) If GðT 1 Þ ¼ GðT 2 Þ and Sð Te 1 Þ ¼ Sð Te 2 Þ then Te 1
ð8Þ
Te 2 . Te 2 .
3. Proposed method Because the activity durations in project networks are usually difficult to estimate or determine exactly, it is reasonable to represent them as linguistic variables or fuzzy numbers. In this paper, the operation time for each activity in the project network is characterized as a positive triangular fuzzy number. The project network
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is represented as an activity-on-node (AON) graph. In addition to the project activities, the AON graph includes two virtual tasks of null duration, the initial node (I) and the ending node (E). In accordance with PERT, the forward pass yields the fuzzy earliest-start and earliest-finish times: e S ei ¼ maxf e S ej d~j g
ð9Þ
j2P ðiÞ
S ei d~i Fe ei ¼ e
ð10Þ
e e is the fuzzy earliest-start time (with e S eI ¼ ð0; 0; 0Þ at the initial node i ¼ I), Fe ei is the fuzzy earliestwhere S i e finish time (with Fe E equal to the project network completion time Te end at the ending node i ¼ EÞ, P ðiÞ is the set of predecessors for activity i, and d~i is the operation time for activity i. The backward pass is performed to calculate the fuzzy latest-start and latest-finish times: Fe li ¼ min f Fe lj Hd~j g
ð11Þ
e S li ¼ Fe li Hd~i
ð12Þ
j2SðiÞ
S li is the fuzzy latest-start time, and SðiÞ is where Fe li is the fuzzy latest-finish time (with Fe lE ¼ Te end when i ¼ E), e the set of successors of activity i. The FPERT implementation employs the generalized mean method [25] to compare fuzzy numbers and compute e S ei , e S li , Fe ei and Fe li for each activity i. In traditional PERT, the float time (slack time) for each activity is either the difference between the latest and earliest starting times or the difference between the latest and earliest finishing times [30]. Once e S ei , e S li , Fe ei l e and F i have been determined for the ith activity, the fuzzy float time is either ~i ¼ e m S li H e S ei
ð13Þ
~ i ¼ Fe li H Fe ei m
ð14Þ
or
~ i is the fuzzy float time, e S li the fuzzy latest-start time, e S ei the fuzzy earliest-start time, Fe li the fuzzy latwhere m e e est-finish time, and F i is the fuzzy earliest-finish time. Property. From the definitions of e S ei , e S li , Fe ei and Fe li , we know that Eqs. (13) and (14) are equivalent, implying that l e l e e S i ¼ Fe i H Fe i . S iHe Proof. Suppose that the a cut of e S ei , e S li , Fe ei , Fe li , and d~i for activity i is given by a a a e S ei ¼ ½seiL ; seiR ; e la ¼ ½sla ; sla ; S
i
iL
iR
a a ea Fe ei ¼ ½fiLe ; fiR ; la la la e F ¼ ½f ; f ;
i
iL
iR
d~ai ¼ ½d aiL ; d aiR ; a a for a 2 ½0; 1. From Eq. (10) we have Fe ei ¼ e S ei d~ai . It follows that a
a
a
a
a
a
e ½fiLe ; fiR ¼ ½seiL ; seiR ½d aiL ; d aiR ¼ ½seiL þ d aiL ; seiR þ d aiR ;
and therefore a
a
fiLe ¼ seiL þ d aiL
and
a
a
e fiR ¼ seiR þ d aiR :
a a Likewise, from Eq. (12), we have e S li ¼ Fe li Hd~ai . It follows that a
a
a
a
a
a
l l ½sliL ; sliR ¼ ½fiLl ; fiR H½d aiL ; d aiR ¼ ½fiLl d aiR ; fiR d aiL ;
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and therefore a
a
sliL ¼ fiLl d aiR
a
a
l sliR ¼ fiR d aiL :
and
a
a
a
a
By adding the equations for fiLe and seiR and the equations for sliL and fieR , we obtain a
a
a
a
l fiLe þ sliR ¼ seiL þ fiR
and
a
a
a
a
a
a
sliL þ fieR ¼ filL þ seiR :
Likewise, by subtracting equations, we get a
a
a
a
l fiLe sliR seiL ¼ fiR
and
a
a
e sliL seiR ¼ fiLl fiR :
Plugging these equations into the definition of fuzzy float time yields a ~ ai ¼ ð e m S li H e S ei Þ a
a
a
a
¼ ½sliL ; sliR H½seiL ; seiR a
a
a
a
¼ ½sliL seiR ; sliR seiL a
a
a
a
e l ¼ ½fiLl fiR ; fiR fiLe a
a
a
a
l e H½fiLe ; fiR ¼ ½fiLl ; fiR a ¼ ð Fe l H Fe e Þ i
i
for a 2 ½0; 1. We conclude that Eqs. (13) and (14) are identical. According to the property, we can easily compute the fuzzy float times of all activities in a project network. In traditional PERT, activity i is said to be a critical activity if its float time is zero. This concept implies that ~ i ¼ ðai ; bi ; ci Þ, then the criticality rises as the fuzzy float time decreases. If the fuzzy float time of activity i is m the criticality of this activity is defined as 8 bi 6 0 > < 1; i CDi ¼ ba ; ai < 0 < bi ð15Þ i ai > : 0; ai P 0 where CDi is the critical degree of activity i. In a project network, a path is a sequence of activities that leads from the initial node to the end node. As determined by the activity criticality, the criticality of an entire path is pðP k Þ ¼ minfCDi g i2P k
ð16Þ
where Pk is the kth path in the network and pðP k Þ is the criticality of the kth path. If path P is critical, then pðP Þ must satisfy pðP Þ ¼ maxk fpðP k Þg. Because fuzzy numbers are used to express the operation times of all activities in the project network, the e we can project network completion time is a fuzzy number, denoted Te end . For a specified required time ð RÞ, e to compute the possibilities within project plans. Suppose that the specified time requirecompare Te end and R e ¼ ðr1 ; r2 ; r3 Þ and the completion time of a project network is Te end ¼ ðe1 ; e2 ; e3 Þ. Then the difference ment is R e is RH e Te end ¼ ðr1 e3 ; r2 e2 ; r3 e1 Þ and the membership function of fuzzy number between Te end and R e Te end is l RH . eR HeT end It follows that the possibility of meeting (PM) a specified project completion time can be determined as follows: e is absolutely larger than Te end , and the project will be completed within the (1) If r1 e3 P 0, then R required time. Therefore, the possibility of meeting the requirement is 100%. e is absolutely smaller than Te end , and the project will not be completed within the (2) If r3 e1 < 0, then R time requirement. Therefore, the possibility of meeting the requirement is zero. e overlap, meaning that the project (3) If r1 e3 < 0 6 r3 e1 , then the membership functions of Te end and R e Te end , the may be completed on time. The larger the positive portion of the membership function RH
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m
1
~
~ Tend
h
d
d
x r1
e3
r2
0
e2
r3 e1
Fig. 4. Membership function of le e . R H T end
higher the possibility of meeting the time requirement. The possibility of meeting (PM) the specified time requirement is 8 r 1 e3 P 0 > < 1; d1 PM ¼ d1 þd2 ; r1 e3 < 0 6 r3 e1 ð17Þ > : 0; r 3 e1 < 0 R R e Te end whered1 ¼ xP0 le e ðxÞdx, d2 ¼ x60 le e ðxÞdx and le e is the membership function of RH R H T end R H T end R H T end (shown in Fig. 4). In a project network, if we shorten the durations of activities on the critical path it will increase the PM value. However, there are often more than two activities that have the same critical degree on the critical path. Under the consideration of resource limitations, the priority for shortening the activity duration must be determined. If the criticality of two activities i and j are identical and CDi ¼ CDj ¼ 1 in a project network, then the generalized mean method [25] can be applied to determine priorities in order to shorten duration. According to the concept of criticality of activity, when the fuzzy float time is smaller the degree of criticality is higher for each activity. Therefore, when the generalized mean value is smaller the priority is to shorten the duration. 4. An example An example is presented in this section to illustrate the proposed method. A project network is given in Fig. 5. The activity operation times are expressed as triangular fuzzy numbers as shown in Table 1. Using the proposed method, the fuzzy earliest-start time, fuzzy earliest-finish time, fuzzy latest-start time and latest-finish time and fuzzy float time of each activity were calculated as shown in Table 1. The activity criticalities were computed based on the fuzzy float times, also shown in Table 1. For example, the criticality of activity 2 was CD2 ¼ 1. The path criticalities for the project network are compared in Table 2 and the path I-2-5-8-9-E was the critical path in this project network.
1
I
3
6
4
7
5
8
9
2
Fig. 5. Graph of project network.
E
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2455
Table 1 Computation results of project network Task
Operation time
Earliest-start time (ES)
Earliest-finish time (EF)
Latest-start time (LS)
Latest-finish time (LF)
Float ~ time ðmÞ
I 1 2 3 4 5 6 7 8 9 E
(0, (1, (2, (2, (2, (3, (3, (4, (4, (3, (0,
(0, 0, 0) (0, 0, 0) (0, 0, 0) (2, 5, 8) (2, 5, 8) (2, 5, 8) (4, 9, 14) (4, 8, 13) (5, 9, 14) (9, 15, 22) (12, 20, 29)
(0, 0, 0) (1, 4, 7) (2, 5, 8) (4, 9, 14) (4, 8, 13) (5, 9, 14) (7, 14, 22) (8, 13, 20) (9, 15, 22) (12, 20, 29) (12, 20, 29)
( 17, 0, 17) (16, 2, 20) (17, 0, 17) (9, 6, 21) (8, 6, 20) (9, 5, 19) (3, 10, 23) (2, 10, 22) (3, 9, 22) (5, 15, 26) (12, 20, 29)
(17, 0, 17) (9, 6, 21) (9, 5, 19) (3, 10, 23) (3, 9, 22) (3, 9, 22) (5, 15, 26) (5, 15, 26) (5, 15, 26) (12, 20, 29) (12, 20, 29)
(17, (16, (17, (17, (16, (17, (17, (15, (17, (17, (17,
0, 4, 5, 4, 3, 4, 5, 5, 6, 5, 0,
0) 7) 8) 6) 5) 6) 8) 7) 8) 7) 0)
0, 2, 0, 1, 1, 0, 1, 2, 0, 0, 0,
Critical degree 17) 20) 17) 19) 18) 17) 19) 18) 17) 17) 17)
1 0.89 1 0.94 0.94 1 0.94 0.88 1 1 1
Table 2 Critical degree of each path No.
Path
pðP k Þ
1 2 3 4 5 6
I-1-3-6-9-E I-2-3-6-9-E I-2-4-6-9-E I-2-4-7-9-E I-2-4-8-9-E I-2-5-8-9-E
0.89 0.94 0.94 0.88 0.94 1
Table 3 PM values with different requirement times e R e1 R e2 R e3 R e4 R e5 R e6 R
Te end ¼ ð8; 9; 11Þ ¼ ð9; 16; 24Þ ¼ ð10; 15; 32Þ ¼ ð17; 25; 33Þ ¼ ð22; 28; 34Þ ¼ ð29; 33; 35Þ
(12, (12, (12, (12, (12, (12,
20, 20, 20, 20, 20, 20,
29) 29) 29) 29) 29) 29)
e Te end RH
PM
(21, 11, 1) (20, 4, 12) (9, 1, 6) (12, 5, 21) (7, 8, 22) (0, 13, 23)
0 0.28 0.41 0.74 0.89 1
When the requirement times and their corresponding PM values are as shown in Table 3, for example, if e 3 ¼ ð10; 15; 32Þ and Te end ¼ ð12; 20; 29Þ then the PM value is 0.41. If the fuzzy requirement time is fixed at R e ¼ ð10; 15; 32Þ, then the activity operation times can be shortened on the critical path to reduce the fuzzy R Table 4 PM values with adjusting the durations of activities on critical path e R
Durations adjustment of nodes
Te end
e Te end RH
(10, 15, 32)
d~2 d~5 d~8 d~9 d~2 d~2 d~2 d~5 d~5 d~8
(9, 18, 28) (10, 19, 29) (10, 19, 29) (12, 19, 26) (11, 17, 25) (10, 17, 26) (9, 17, 25) (10, 19, 29) (10, 18, 26) (10, 18, 26)
(18, (19, (19, (16, (15, (16, (15, (19, (16, (16,
¼ ð2; 3; 5Þ ¼ ð3; 3; 3Þ ¼ ð4; 4; 5Þ ¼ ð3; 4; 4Þ ¼ ð2; 3; 5Þ ¼ ð2; 3; 5Þ ¼ ð2; 3; 5Þ ¼ ð3; 3; 3Þ ¼ ð3; 3; 3Þ ¼ ð4; 4; 5Þ
and and and and and and
d~5 d~8 d~9 d~8 d~9 d~9
¼ ð3; 3; 3Þ ¼ ð4; 4; 5Þ ¼ ð3; 4; 4Þ ¼ ð4; 4; 5Þ ¼ ð3; 4; 4Þ ¼ ð3; 4; 4Þ
3, 23) 4, 22) 4, 22) 4, 20) 2, 21) 2, 22) 8, 23) 4, 22) 3, 22) 3, 22)
PM 0.49 0.46 0.46 0.46 0.53 0.53 0.74 0.46 0.51 0.51
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project network completion time. Because the critical path was I-2-5-8-9-E in the network, the operation times for activities 2, 5, 8 and 9 would be shortened to increase the PM value as shown in Table 4. For example, when the operation time of activity 2 from (2, 5, 8) was shortened to (2, 3, 5) the fuzzy completion time became Te end ¼ ð9; 18; 28Þ and the corresponding PM value was 0.49. When the operation times for activities 2 and 9 were shortened at the same time (shown as Table 4), the corresponding PM value was 0.74. As a result of these calculations, the PM value was increased to 74% for e ¼ ð10; 15; 32Þ. It is obviously easy to shorten the critical activity operation the fixed fuzzy requirement time R times to increase the PM of the entire project by using the proposed method. Determining activity criticality and computing the PM becomes a simple job in a project network. 5. Results of comparison In order to validate the effectiveness of the proposed method, the proposed method, the method of Dubois et al. [12] and the method of Mon et al. [26] were compared by finding the critical path based on the data in Table 1. According to the method of Dubois et al. [12], the activity operation times can be represented as fuzzy numbers. The crisp duration of each activity can be selected randomly within the range of the fuzzy operation time. The critical path can then be determined under different combinations of crisp durations for all activities in a project network. Based on the data in Table 1, three combinations ðX1 , X2 and X3 Þ are applied to determine the critical path. For the first combination X1 ¼ ð7; 8; 6; 5; 6; 8; 7; 8; 7Þ, two critical paths were found: I-23-6-9-E and I-2-5-8-9-E, shown in Table 5. Likewise, the critical paths I-2-5-8-9-E and I-1-3-6-9-E for the second and third combinations X2 and X3 are shown in Tables 6 and 7. Clearly, different combinations of activity durations yield different critical paths in a project network. In the method of Dubois et al. [12], the activity criticalities and the critical path in a project network could not easily be determined. According to the method of Mon et al. [26], the critical paths with different a and k values are shown in Table 8. Obviously, different combinations of a and k values may yield different critical paths in a project network in the method of Mon et al. [26]. The comparison results demonstrate that the method proposed in this paper is more effective in determining the activity criticalities and finding the critical path. These are very important contributions of
Table 5 Calculation result of X1 Task
I
1
2
3
4
5
6
7
8
9
E
Operation time Earliest start time (ES) Earliest finish time (EF) Latest finish time (LF) Latest start time (LS) Float time (m)
0 0 0 0 0 0
7 0 7 8 1 1
8 0 8 8 0 0
6 8 14 14 8 0
5 8 13 14 9 1
6 8 14 14 8 0
8 14 22 22 14 0
7 13 20 22 15 2
8 14 22 22 14 0
7 22 29 29 22 0
0 29 29 29 29 0
Critical path I Critical path II
I-2-3-6-9 E I-2-5-8-9-E
Table 6 Calculation result of X2 Task
I
1
2
3
4
5
6
7
8
9
E
Operation time Earliest start time (ES) Earliest finish time (EF) Latest finish time (LF) Latest start time (LS) Float time (m)
0 0 0 0 0 0
4 0 4 6 2 2
5 0 5 5 0 0
4 5 9 10 6 1
3 5 8 9 6 1
4 5 9 9 5 0
5 9 14 15 10 1
5 8 13 15 10 2
6 9 15 15 9 0
5 15 20 20 15 0
0 20 20 20 20 0
Critical path
I-2-5-8-9-E
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Table 7 Calculation result of X3 Task
I
1
2
3
4
5
6
7
8
9
E
Operation time Earliest start time (ES) Earliest finish time (EF) Latest finish time( LF) Latest start time (LS) Float time (m)
0 0 0 0 0 0
7 0 7 7 0 0
2 0 2 7 5 5
6 7 13 13 7 0
5 2 7 13 8 6
3 2 5 17 14 12
8 13 21 21 13 0
7 7 14 21 14 7
4 7 11 21 17 10
3 21 24 24 21 0
0 24 24 24 24 0
Critical path
I-1-3-6-9-E
Table 8 Critical paths with different a and k values a
k
Critical path
a¼0
k¼0 k ¼ 0:5 k¼1
a ¼ 0:5
k¼0 k ¼ 0:5 k¼1 –
I-2-5-8-9-E I-2-5-8-9-E I-2-5-8-9-E I-2-3-6-9-E I-2-5-8-9-E I-2-5-8-9-E I-2-5-8-9-E I-2-5-8-9-E
a¼1
this paper for solving the fuzzy PERT problems. Besides, a computer program has been developed to deal with the larger number of arguments (fuzzy numbers). Using this computer program, managers can determine the activity criticalities and find the critical path more easily and efficiently. 6. Conclusions PERT (program evaluation and review technique) is the most widely used technique for planning and coordinating large-scale projects. The main assumption in PERT is that the activity durations in a project can be estimated precisely and that they are statistically independent. As a result, in cases where this assumption does not hold, PERT appears to lead to poor estimation and inadequate management responses. In this study, FPERT methods were employed to overcome this problem. Triangular fuzzy numbers were used to represent the activity durations in a project network. Fuzzy bounds were computed for the starting and finishing times in each activity. A critical degree index based on the fuzzy float time was defined to calculate the critical degree of each activity and path. A second index was used to analyze the possibility of meeting a specified required time for a project network in an unpredictable environment. Using the proposed model, information on the status of the project network is immediately available and sensitivity analysis can be conducted by varying the durations of activities on the critical path. Acknowledgements The authors gratefully acknowledge the financial support of Taiwan’s National Science Council (Project number NSC 92-2213-E-212-008). The authors also express appreciation to Ms. C. Troy and Mr. Philip du Plessis for their editorial assistance. References [1] S. Avraham, Project segmentation – a tool for project management, International Journal of Project Management 15 (1997) 15–19. [2] A. Azaron, C. Perkgoz, M. Sakawa, A genetic algorithm approach for the time-cost trade-off in PERT networks, Applied Mathematics and Computation 168 (2005) 1317–1339.
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[3] A. Azaron, H. Katagiri, M. Sakawa, K. Kato, A. Memariani, A multi-objective resource allocation problem in PERT networks, European Journal of Operational Research 172 (2006) 838–854. [4] S. Chanas, J. Kamburowski, The use of fuzzy variables in PERT, Fuzzy Sets and Systems 5 (1981) 11–19. [5] S. Chanas, P. Zielinski, Critical path analysis in the network with fuzzy task times, Fuzzy Sets and Systems 122 (2001) 195–204. [6] S.M. Chen, T.H. Chang, Finding multiple possible critical paths using fuzzy PERT, IEEE Transactions on Systems, man and Cybernetics – Part B: Cybernetics 31 (2001) 930–937. [7] S.J. Chen, C.L. Hwang, Fuzzy Multiple Attribute Decision Making-Methods and Applications, Springer-Verlag, Berlin Heidelberg, 1992. [8] C.W. Chiu, H.L. Ping, C.T. Yingn, Resource-constrained project management using enhanced theory of constraint, International Journal of Project Management 20 (2002) 561–567. [9] C.W. Dawson, R.J. Dawson, Generalised task-on-the-node networks for managing uncertainty in projects, International Journal of Project Management 13 (1995) 353–362. [10] R.J. Dawson, R.J. Dawson, Practical proposals for managing uncertainty and risk in project planning, International Journal of Project Management 16 (1998) 299–310. [11] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academy Press, 1980. [12] D. Dubois, H. Fargier, V. Galvagonon, On latest starting times and floats in task networks with ill-known durations, European Journal of Operational Research 147 (2003) 266–280. [13] H. Fargier, V. Galvagnon, Fuzzy PERT in series-parallel graphs, in: Prooceedings of Nineth International Conference on Fuzzy Systems, 2000, pp. 717–722. [14] S.M.T. Fatemi Ghomi, E. Teimouri, Path critical index and task critical index in PERT networks, European Journal of Operational Research 141 (2002) 147–152. [15] S.M.T. Fatemi Ghomi, M. Rabbani, A new structural mechanism for reducibility of stochastic PERT networks, European Journal of Operational Research 145 (2003) 394–402. [16] K.R. Graham, Critical chain: the theory of constraints applied to project management, International Journal of Project Management 18 (2000) 173–177. [17] M. Giovanni, Measuring uncertainty and criticality in network planning by PERT-path technique, International Journal of Project Management 15 (1997) 377–387. [18] M. Hapke, R. Slowinski, Fuzzy project scheduling system for software development, Fuzzy Sets and Systems 67 (1994) 101–107. [19] M. Hapke, R. Slowinski, Fuzzy priority heuristics for project scheduling, Fuzzy Sets and Systems 83 (1996) 291–299. [20] J. Heizer, B. Render, Principles of Operations Management, Prentice-Hall, 1999. [21] K.N. Jha, K.C. Iyer, Critical determinants of project coordination, International Journal of Project Management 24 (2006) 314–322. [22] A. Kaufmann, M.M. Gupta, Introduction to Fuzzy Arithmetic: Theory and Applications, International Thomson Computer Press, London, 1991. [23] G.J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall, International Inc., 1995. [24] D. Kuchta, Use of fuzzy numbers in project risk (criticality) assessment, International Journal of Project Management 19 (2001) 305– 310. [25] E.S. Lee, R.J. Li, Comparison of fuzzy numbers based on the probability measure of fuzzy events, Computers and Mathematics with Applications 15 (1988) 887–896. [26] D.L. Mon, C.H. Cheng, H.C. Lu, Application of fuzzy distributions on project management, Fuzzy Sets Systems 73 (1995) 227–234. [27] H. Neil, The prediction and control of project duration: a recursive model, International Journal of Project Management 19 (2001) 401–409. [28] I.M. Premachandrak, An approximation of the task duration distribution in PERT, Computers and Operations Research 28 (2001) 443–452. [29] G. Roy, P. Nava, S. Israel, Integrating system analysis and project management tools, International Journal of Project Management 20 (2002) 461–468. [30] W.J. Stevenson, Operation Management, seventh ed., McGraw-Hill, 2002. [31] W.C. Wang, Impact of soft logic on the probabilistic duration of construction projects, International Journal of Project Management 23 (2005) 600–610. [32] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353. [33] L.A. Zadeh, Toward a generalized theory of uncertainty (GTU) – an outline, Information Sciences 17 (2005) 1–40. [34] H.J. Zimmerman, Fuzzy Set Theory and Its Applications, second ed., Kluwer Academic Publishers, Boston, 1991.
Applying fuzzy method for measuring criticality in project network Chen-Tung Chen a
a,*
, Sue-Fen Huang
b
Department of Information Management, National United University 1, Lien Da, Kung-Ching Li, Miao-Li 36003, Taiwan b Department of Information Management, Da-Yeh University 112, Shan-Jiau Road, Da-Tsuen, Changhua, Taiwan Received 24 January 2007; accepted 25 January 2007
Abstract Program evaluation and review technique (PERT) is widely used as a tool for managing large-scale projects. In the traditional PERT model, the durations of activities (tasks) are either represented as crisp numbers or drawn from the beta distribution to estimate the task durations such as pessimistic, most likely and optimistic times. However, the operation time for each activity is usually difficult to define and estimate precisely in a real situation. The aim of this paper is to present an analytical method for measuring the criticality in a project network with fuzzy activity times. Triangular fuzzy numbers are used to express the operation times for all activities in a project network. A new model that combines fuzzy set theory with the PERT technique is proposed to determine the critical degrees of activities (tasks) and paths. In the proposed model, a possibility index is defined to identify the likelihood of meeting a specified required time for a project network. At the end of the paper, an example is presented to compare with those obtained using the proposed method as well as other methods. The comparisons reveal that the method proposed in this paper is more effective in determining the activity criticalities and finding the critical path. Ó 2007 Published by Elsevier Inc. Keywords: Project management; Fuzzy set; Program evaluation and review technique (PERT); Fuzzy PERT
1. Introduction In recent years, the range of project management applications has greatly expanded. Project management concerns the scheduling and control of activities (tasks) in such a way that the project can be completed in as little time as possible [1,29]. To ensure the project’s success, the project management team must identify the stakeholders, determine their needs and expectations, and manage those needs and expectations. A project network is defined as a set of activities that must be performed according to precedence constraints stating which activities must start after the completion of specified other activities [12]. Such a project network can be represented as a directed graph. There are two slightly different conventions for representing these network *
Corresponding author. Tel.: +886 37 381833; fax: +886 37 330776. E-mail address: [email protected] (C.-T. Chen).
0020-0255/$ - see front matter Ó 2007 Published by Elsevier Inc. doi:10.1016/j.ins.2007.01.035
C.-T. Chen, S.-F. Huang / Information Sciences 177 (2007) 2448–2458
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graphs, the activity-on-arrow (AOA) graph and the activity-on-node (AON) graph [30]. These conventions are illustrated in Figs. 1 and 2. In the AON graph, the nodes represent activities and the arcs represent precedence relations. A path through a project network is one of the routes from the starting node to the ending node. The length of a path is the sum of the durations of the activities on the path. The project duration equals the length of the longest path through the project network. The longest path is called the critical path in the network. A project is deemed complete if work along all paths is complete. After the durations and precedence relations of activities have been determined, project management techniques are used to calculate the completion time of the project. The most commonly used project management techniques are the Grant Chart, Milestone, Critical Path Management (CPM), and Project Evaluation and Review Technique (PERT) [8,27,31]. PERT is the most widely used management technique for planning and coordinating large-scale projects [2,6,11,12,14,16– 19,21,31]. It was developed in the 1950s to help managers schedule, monitor, and control large and complex projects [20]. Nowadays, it is extensively used both in industries and service organizations. By using PERT, managers are able to obtain [3,30]: 1. 2. 3. 4.
A graphical display of project activities (tasks). An estimate of how long the project will take. An indication of which activities are the most critical for timely project completion. An indication of how long any task can be delayed without delaying the project.
In traditional PERT techniques, various dynamic activity durations must be represented either as crisp numbers or as random variables drawn from certain probability distributions (normally, a Beta distribution). In PERT, we must specify three time estimates for each activity: an optimistic time, a most probable (or most likely) time, and a pessimistic time. The optimistic time is the fastest possible activity completion time for the given activity. The probable time is the time that will most likely be required. The pessimistic time is the ‘‘worst’’ task time that could be expected. The three time estimates are used to calculate both the expected completion time and variance for each activity [3,9,10,28]. The main drawback of PERT technique is the difficulty of obtaining the time estimates. It is hard to get precise information about activity durations in some situations, such as early rough planning in long range projects [21]. In a real world situation, operation times of activities in a project network may be difficult to define and estimate exactly. Therefore, it is important to compute the variance of the project completion time in a network [2,3]. In recent years many researchers have combined fuzzy-set theory with PERT to handle time estimates in project planning and control problems. The result is a new approach called fuzzy PERT (FPERT) [6,12– 15,18,19,24–26]. FPERT was first presented by Chanas and Kamburowski [4], who used fuzzy numbers to
1
a 3
4 c
2
b
Fig. 1. Activity on arc.
a c b
Fig. 2. Activity on node.
d
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represent activity durations in project networks. The possibility distribution of the project completion time is derived from the individual activity duration possibility distributions once they are known. All fuzzy activity durations are calculated based on the three time estimates. The fuzzy a-cut is then applied to calculate the upper and lower bounds on the completion time of the project network. One problem in Chanas and Kamburowski is that different values of a result in different bounds on the completion time. In this situation, there is no effective way to indicate the critical activities and paths for a project network. Mon et al. [26] remedied this problem by assuming that the duration of each activity is a e a ¼ ½aa ; aa , where positive fuzzy number. Using the a-cut of each fuzzy duration, they denoted the interval A L R a a aL and aR were the lower and upper duration bounds. Using traditional PERT, a linear combination of the duration bounds would be used to represent the operation time of each activity and determine the critical activities and paths. However, the a values would yield different critical activities and paths in this approach. Chanas and Zielinski [5] assume that the operation time of each activity can be represented as a crisp value, interval or a fuzzy number. The calculation procedure of Mon et al. [26] is similar. The method proposed in Dubois et al. [12] assigns a different level of importance to each activity on a critical path for a randomly chosen set of activities. All of these FPERT methods reduce uncertainties by using fuzzy numbers to represent the activity operation times. However, the procedures for calculating the critical degree of each activity and path are quite complicated. Moreover, no effective and straightforward method is available to estimate the possibility of meeting a specified requirement time. In fact, uncertainty is an attribute of information [33]. The subjective judgment or prediction of an expert for the times of activities may be ill defined. It is reasonable to represent them as linguistic variables or fuzzy numbers [33]. In this paper, a FPERT method is proposed to deal with completion time management and the critical degrees of all activities for a project network. First, the fuzzy completion time of a project network was derived from the fuzzy activity operation times. Second, an index to compute the critical degrees of activities and paths was employed. Then, a straightforward approach to identify the critical activities and critical paths in a project network was used. Finally, an index that determines the possibility of a project meeting a specified requirement time was developed. 2. Fuzzy sets and notations A fuzzy set can be mathematically constructed by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set [32,33]. This grade corresponds to the individual’s similarity to the concept represented by the fuzzy set. Thus, individuals may belong in the fuzzy set to a greater or lesser degree as indicated by a larger or smaller membership grade. As already mentioned, these membership grades are often represented by real numbers ranging from a minimum of 0 to a maximum of 1 [23,33]. e is a fuzzy set whose membership function l ðxÞ satisfies the following conditions [23]: The fuzzy number A e A
1. 2. 3.
leðxÞ is piecewise continuous; A leðxÞ is a convex fuzzy subset; A leðxÞ is the normality of a fuzzy subset, implying that for at least one element x0 the membership grade A must be 1, i.e. leðx0 Þ ¼ 1. A
e is defined as The a-cut of a fuzzy number A e a ¼ fxi : l ðxi Þ P a; xi 2 X g A eA
ð1Þ
where a 2 ½0; 1. e a ¼ ½aa ; aa , where aa e a represents a non-empty bounded interval contained in X, denoted A The symbol A L R L and aaR are the lower and upper bounds of the closed interval [22,34]. ~ and ~n are m ~ a ¼ ½mal ; mau and ~na ¼ ½nal ; nau (a 2 ½0; 1Þ. Based The a-cut of any two positive fuzzy numbers m on the interval of confidence [22], the following major operations can be applied to two positive fuzzy numbers ~ and ~ m n:
C.-T. Chen, S.-F. Huang / Information Sciences 177 (2007) 2448–2458
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µ~ T (x )
1
0
l
m
u
x
Fig. 3. Positive triangular fuzzy number Te .
a
~ ðmðþÞ~ nÞ ¼ ½mal þ nal ; mau þ nau ~ ðmðÞ~ nÞa ¼ ½mal nau ; mau nal
ð2Þ ð3Þ
Particularly common in practical applications are triangular fuzzy numbers, denoted Te ¼ ðl; m; uÞ. When l > 0, Te is a positive triangular fuzzy number (PTFN) [11,34]. Fig. 3 shows the membership function of a positive triangular fuzzy number Te , defined as 8 xl > < ml ; l 6 x 6 m ux le ðxÞ ¼ um ð4Þ ; m6x6u T > : 0; otherwise where l > 0. Given two PTFNs Te 1 ¼ ðl1 ; m1 ; u1 Þ and Te 2 ¼ ðl2 ; m2 ; u2 Þ, fuzzy addition and subtraction can be performed as follows [22]: Te 1 Te 2 ¼ ðl1 þ l2 ; m1 þ m2 ; u1 þ u2 Þ Te 1 H Te 2 ¼ ðl1 u2 ; m1 m2 ; u1 l2 Þ
ð5Þ ð6Þ
Many ranking methods have been developed to transform fuzzy numbers into crisp values [7]. Lee and Li [25] presented the generalized mean value method, an effective method for ranking and comparing fuzzy numbers. For a triangular fuzzy number Te ¼ ðl; m; uÞ, the generalized mean value Gð Te Þ and deviation Sð Te Þ are given by [25] Gð Te Þ ¼
lþmþu 3
ð7Þ
and 1 2 ½l þ m2 þ u2 lm lu mu 18 Two triangular fuzzy numbers Te 1 ¼ ðl1 ; m1 ; u1 Þ and Te 2 ¼ ðl2 ; m2 ; u2 Þ can be compared as follows: Sð Te Þ ¼
(1) If GðT 1 Þ > GðT 2 Þ then Te 1 > Te 2 . (2) If GðT 1 Þ ¼ GðT 2 Þ and Sð Te 1 Þ < Sð Te 2 Þ then Te 1 > (3) If GðT 1 Þ ¼ GðT 2 Þ and Sð Te 1 Þ ¼ Sð Te 2 Þ then Te 1
ð8Þ
Te 2 . Te 2 .
3. Proposed method Because the activity durations in project networks are usually difficult to estimate or determine exactly, it is reasonable to represent them as linguistic variables or fuzzy numbers. In this paper, the operation time for each activity in the project network is characterized as a positive triangular fuzzy number. The project network
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is represented as an activity-on-node (AON) graph. In addition to the project activities, the AON graph includes two virtual tasks of null duration, the initial node (I) and the ending node (E). In accordance with PERT, the forward pass yields the fuzzy earliest-start and earliest-finish times: e S ei ¼ maxf e S ej d~j g
ð9Þ
j2P ðiÞ
S ei d~i Fe ei ¼ e
ð10Þ
e e is the fuzzy earliest-start time (with e S eI ¼ ð0; 0; 0Þ at the initial node i ¼ I), Fe ei is the fuzzy earliestwhere S i e finish time (with Fe E equal to the project network completion time Te end at the ending node i ¼ EÞ, P ðiÞ is the set of predecessors for activity i, and d~i is the operation time for activity i. The backward pass is performed to calculate the fuzzy latest-start and latest-finish times: Fe li ¼ min f Fe lj Hd~j g
ð11Þ
e S li ¼ Fe li Hd~i
ð12Þ
j2SðiÞ
S li is the fuzzy latest-start time, and SðiÞ is where Fe li is the fuzzy latest-finish time (with Fe lE ¼ Te end when i ¼ E), e the set of successors of activity i. The FPERT implementation employs the generalized mean method [25] to compare fuzzy numbers and compute e S ei , e S li , Fe ei and Fe li for each activity i. In traditional PERT, the float time (slack time) for each activity is either the difference between the latest and earliest starting times or the difference between the latest and earliest finishing times [30]. Once e S ei , e S li , Fe ei l e and F i have been determined for the ith activity, the fuzzy float time is either ~i ¼ e m S li H e S ei
ð13Þ
~ i ¼ Fe li H Fe ei m
ð14Þ
or
~ i is the fuzzy float time, e S li the fuzzy latest-start time, e S ei the fuzzy earliest-start time, Fe li the fuzzy latwhere m e e est-finish time, and F i is the fuzzy earliest-finish time. Property. From the definitions of e S ei , e S li , Fe ei and Fe li , we know that Eqs. (13) and (14) are equivalent, implying that l e l e e S i ¼ Fe i H Fe i . S iHe Proof. Suppose that the a cut of e S ei , e S li , Fe ei , Fe li , and d~i for activity i is given by a a a e S ei ¼ ½seiL ; seiR ; e la ¼ ½sla ; sla ; S
i
iL
iR
a a ea Fe ei ¼ ½fiLe ; fiR ; la la la e F ¼ ½f ; f ;
i
iL
iR
d~ai ¼ ½d aiL ; d aiR ; a a for a 2 ½0; 1. From Eq. (10) we have Fe ei ¼ e S ei d~ai . It follows that a
a
a
a
a
a
e ½fiLe ; fiR ¼ ½seiL ; seiR ½d aiL ; d aiR ¼ ½seiL þ d aiL ; seiR þ d aiR ;
and therefore a
a
fiLe ¼ seiL þ d aiL
and
a
a
e fiR ¼ seiR þ d aiR :
a a Likewise, from Eq. (12), we have e S li ¼ Fe li Hd~ai . It follows that a
a
a
a
a
a
l l ½sliL ; sliR ¼ ½fiLl ; fiR H½d aiL ; d aiR ¼ ½fiLl d aiR ; fiR d aiL ;
C.-T. Chen, S.-F. Huang / Information Sciences 177 (2007) 2448–2458
2453
and therefore a
a
sliL ¼ fiLl d aiR
a
a
l sliR ¼ fiR d aiL :
and
a
a
a
a
By adding the equations for fiLe and seiR and the equations for sliL and fieR , we obtain a
a
a
a
l fiLe þ sliR ¼ seiL þ fiR
and
a
a
a
a
a
a
sliL þ fieR ¼ filL þ seiR :
Likewise, by subtracting equations, we get a
a
a
a
l fiLe sliR seiL ¼ fiR
and
a
a
e sliL seiR ¼ fiLl fiR :
Plugging these equations into the definition of fuzzy float time yields a ~ ai ¼ ð e m S li H e S ei Þ a
a
a
a
¼ ½sliL ; sliR H½seiL ; seiR a
a
a
a
¼ ½sliL seiR ; sliR seiL a
a
a
a
e l ¼ ½fiLl fiR ; fiR fiLe a
a
a
a
l e H½fiLe ; fiR ¼ ½fiLl ; fiR a ¼ ð Fe l H Fe e Þ i
i
for a 2 ½0; 1. We conclude that Eqs. (13) and (14) are identical. According to the property, we can easily compute the fuzzy float times of all activities in a project network. In traditional PERT, activity i is said to be a critical activity if its float time is zero. This concept implies that ~ i ¼ ðai ; bi ; ci Þ, then the criticality rises as the fuzzy float time decreases. If the fuzzy float time of activity i is m the criticality of this activity is defined as 8 bi 6 0 > < 1; i CDi ¼ ba ; ai < 0 < bi ð15Þ i ai > : 0; ai P 0 where CDi is the critical degree of activity i. In a project network, a path is a sequence of activities that leads from the initial node to the end node. As determined by the activity criticality, the criticality of an entire path is pðP k Þ ¼ minfCDi g i2P k
ð16Þ
where Pk is the kth path in the network and pðP k Þ is the criticality of the kth path. If path P is critical, then pðP Þ must satisfy pðP Þ ¼ maxk fpðP k Þg. Because fuzzy numbers are used to express the operation times of all activities in the project network, the e we can project network completion time is a fuzzy number, denoted Te end . For a specified required time ð RÞ, e to compute the possibilities within project plans. Suppose that the specified time requirecompare Te end and R e ¼ ðr1 ; r2 ; r3 Þ and the completion time of a project network is Te end ¼ ðe1 ; e2 ; e3 Þ. Then the difference ment is R e is RH e Te end ¼ ðr1 e3 ; r2 e2 ; r3 e1 Þ and the membership function of fuzzy number between Te end and R e Te end is l RH . eR HeT end It follows that the possibility of meeting (PM) a specified project completion time can be determined as follows: e is absolutely larger than Te end , and the project will be completed within the (1) If r1 e3 P 0, then R required time. Therefore, the possibility of meeting the requirement is 100%. e is absolutely smaller than Te end , and the project will not be completed within the (2) If r3 e1 < 0, then R time requirement. Therefore, the possibility of meeting the requirement is zero. e overlap, meaning that the project (3) If r1 e3 < 0 6 r3 e1 , then the membership functions of Te end and R e Te end , the may be completed on time. The larger the positive portion of the membership function RH
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C.-T. Chen, S.-F. Huang / Information Sciences 177 (2007) 2448–2458
m
1
~
~ Tend
h
d
d
x r1
e3
r2
0
e2
r3 e1
Fig. 4. Membership function of le e . R H T end
higher the possibility of meeting the time requirement. The possibility of meeting (PM) the specified time requirement is 8 r 1 e3 P 0 > < 1; d1 PM ¼ d1 þd2 ; r1 e3 < 0 6 r3 e1 ð17Þ > : 0; r 3 e1 < 0 R R e Te end whered1 ¼ xP0 le e ðxÞdx, d2 ¼ x60 le e ðxÞdx and le e is the membership function of RH R H T end R H T end R H T end (shown in Fig. 4). In a project network, if we shorten the durations of activities on the critical path it will increase the PM value. However, there are often more than two activities that have the same critical degree on the critical path. Under the consideration of resource limitations, the priority for shortening the activity duration must be determined. If the criticality of two activities i and j are identical and CDi ¼ CDj ¼ 1 in a project network, then the generalized mean method [25] can be applied to determine priorities in order to shorten duration. According to the concept of criticality of activity, when the fuzzy float time is smaller the degree of criticality is higher for each activity. Therefore, when the generalized mean value is smaller the priority is to shorten the duration. 4. An example An example is presented in this section to illustrate the proposed method. A project network is given in Fig. 5. The activity operation times are expressed as triangular fuzzy numbers as shown in Table 1. Using the proposed method, the fuzzy earliest-start time, fuzzy earliest-finish time, fuzzy latest-start time and latest-finish time and fuzzy float time of each activity were calculated as shown in Table 1. The activity criticalities were computed based on the fuzzy float times, also shown in Table 1. For example, the criticality of activity 2 was CD2 ¼ 1. The path criticalities for the project network are compared in Table 2 and the path I-2-5-8-9-E was the critical path in this project network.
1
I
3
6
4
7
5
8
9
2
Fig. 5. Graph of project network.
E
C.-T. Chen, S.-F. Huang / Information Sciences 177 (2007) 2448–2458
2455
Table 1 Computation results of project network Task
Operation time
Earliest-start time (ES)
Earliest-finish time (EF)
Latest-start time (LS)
Latest-finish time (LF)
Float ~ time ðmÞ
I 1 2 3 4 5 6 7 8 9 E
(0, (1, (2, (2, (2, (3, (3, (4, (4, (3, (0,
(0, 0, 0) (0, 0, 0) (0, 0, 0) (2, 5, 8) (2, 5, 8) (2, 5, 8) (4, 9, 14) (4, 8, 13) (5, 9, 14) (9, 15, 22) (12, 20, 29)
(0, 0, 0) (1, 4, 7) (2, 5, 8) (4, 9, 14) (4, 8, 13) (5, 9, 14) (7, 14, 22) (8, 13, 20) (9, 15, 22) (12, 20, 29) (12, 20, 29)
( 17, 0, 17) (16, 2, 20) (17, 0, 17) (9, 6, 21) (8, 6, 20) (9, 5, 19) (3, 10, 23) (2, 10, 22) (3, 9, 22) (5, 15, 26) (12, 20, 29)
(17, 0, 17) (9, 6, 21) (9, 5, 19) (3, 10, 23) (3, 9, 22) (3, 9, 22) (5, 15, 26) (5, 15, 26) (5, 15, 26) (12, 20, 29) (12, 20, 29)
(17, (16, (17, (17, (16, (17, (17, (15, (17, (17, (17,
0, 4, 5, 4, 3, 4, 5, 5, 6, 5, 0,
0) 7) 8) 6) 5) 6) 8) 7) 8) 7) 0)
0, 2, 0, 1, 1, 0, 1, 2, 0, 0, 0,
Critical degree 17) 20) 17) 19) 18) 17) 19) 18) 17) 17) 17)
1 0.89 1 0.94 0.94 1 0.94 0.88 1 1 1
Table 2 Critical degree of each path No.
Path
pðP k Þ
1 2 3 4 5 6
I-1-3-6-9-E I-2-3-6-9-E I-2-4-6-9-E I-2-4-7-9-E I-2-4-8-9-E I-2-5-8-9-E
0.89 0.94 0.94 0.88 0.94 1
Table 3 PM values with different requirement times e R e1 R e2 R e3 R e4 R e5 R e6 R
Te end ¼ ð8; 9; 11Þ ¼ ð9; 16; 24Þ ¼ ð10; 15; 32Þ ¼ ð17; 25; 33Þ ¼ ð22; 28; 34Þ ¼ ð29; 33; 35Þ
(12, (12, (12, (12, (12, (12,
20, 20, 20, 20, 20, 20,
29) 29) 29) 29) 29) 29)
e Te end RH
PM
(21, 11, 1) (20, 4, 12) (9, 1, 6) (12, 5, 21) (7, 8, 22) (0, 13, 23)
0 0.28 0.41 0.74 0.89 1
When the requirement times and their corresponding PM values are as shown in Table 3, for example, if e 3 ¼ ð10; 15; 32Þ and Te end ¼ ð12; 20; 29Þ then the PM value is 0.41. If the fuzzy requirement time is fixed at R e ¼ ð10; 15; 32Þ, then the activity operation times can be shortened on the critical path to reduce the fuzzy R Table 4 PM values with adjusting the durations of activities on critical path e R
Durations adjustment of nodes
Te end
e Te end RH
(10, 15, 32)
d~2 d~5 d~8 d~9 d~2 d~2 d~2 d~5 d~5 d~8
(9, 18, 28) (10, 19, 29) (10, 19, 29) (12, 19, 26) (11, 17, 25) (10, 17, 26) (9, 17, 25) (10, 19, 29) (10, 18, 26) (10, 18, 26)
(18, (19, (19, (16, (15, (16, (15, (19, (16, (16,
¼ ð2; 3; 5Þ ¼ ð3; 3; 3Þ ¼ ð4; 4; 5Þ ¼ ð3; 4; 4Þ ¼ ð2; 3; 5Þ ¼ ð2; 3; 5Þ ¼ ð2; 3; 5Þ ¼ ð3; 3; 3Þ ¼ ð3; 3; 3Þ ¼ ð4; 4; 5Þ
and and and and and and
d~5 d~8 d~9 d~8 d~9 d~9
¼ ð3; 3; 3Þ ¼ ð4; 4; 5Þ ¼ ð3; 4; 4Þ ¼ ð4; 4; 5Þ ¼ ð3; 4; 4Þ ¼ ð3; 4; 4Þ
3, 23) 4, 22) 4, 22) 4, 20) 2, 21) 2, 22) 8, 23) 4, 22) 3, 22) 3, 22)
PM 0.49 0.46 0.46 0.46 0.53 0.53 0.74 0.46 0.51 0.51
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project network completion time. Because the critical path was I-2-5-8-9-E in the network, the operation times for activities 2, 5, 8 and 9 would be shortened to increase the PM value as shown in Table 4. For example, when the operation time of activity 2 from (2, 5, 8) was shortened to (2, 3, 5) the fuzzy completion time became Te end ¼ ð9; 18; 28Þ and the corresponding PM value was 0.49. When the operation times for activities 2 and 9 were shortened at the same time (shown as Table 4), the corresponding PM value was 0.74. As a result of these calculations, the PM value was increased to 74% for e ¼ ð10; 15; 32Þ. It is obviously easy to shorten the critical activity operation the fixed fuzzy requirement time R times to increase the PM of the entire project by using the proposed method. Determining activity criticality and computing the PM becomes a simple job in a project network. 5. Results of comparison In order to validate the effectiveness of the proposed method, the proposed method, the method of Dubois et al. [12] and the method of Mon et al. [26] were compared by finding the critical path based on the data in Table 1. According to the method of Dubois et al. [12], the activity operation times can be represented as fuzzy numbers. The crisp duration of each activity can be selected randomly within the range of the fuzzy operation time. The critical path can then be determined under different combinations of crisp durations for all activities in a project network. Based on the data in Table 1, three combinations ðX1 , X2 and X3 Þ are applied to determine the critical path. For the first combination X1 ¼ ð7; 8; 6; 5; 6; 8; 7; 8; 7Þ, two critical paths were found: I-23-6-9-E and I-2-5-8-9-E, shown in Table 5. Likewise, the critical paths I-2-5-8-9-E and I-1-3-6-9-E for the second and third combinations X2 and X3 are shown in Tables 6 and 7. Clearly, different combinations of activity durations yield different critical paths in a project network. In the method of Dubois et al. [12], the activity criticalities and the critical path in a project network could not easily be determined. According to the method of Mon et al. [26], the critical paths with different a and k values are shown in Table 8. Obviously, different combinations of a and k values may yield different critical paths in a project network in the method of Mon et al. [26]. The comparison results demonstrate that the method proposed in this paper is more effective in determining the activity criticalities and finding the critical path. These are very important contributions of
Table 5 Calculation result of X1 Task
I
1
2
3
4
5
6
7
8
9
E
Operation time Earliest start time (ES) Earliest finish time (EF) Latest finish time (LF) Latest start time (LS) Float time (m)
0 0 0 0 0 0
7 0 7 8 1 1
8 0 8 8 0 0
6 8 14 14 8 0
5 8 13 14 9 1
6 8 14 14 8 0
8 14 22 22 14 0
7 13 20 22 15 2
8 14 22 22 14 0
7 22 29 29 22 0
0 29 29 29 29 0
Critical path I Critical path II
I-2-3-6-9 E I-2-5-8-9-E
Table 6 Calculation result of X2 Task
I
1
2
3
4
5
6
7
8
9
E
Operation time Earliest start time (ES) Earliest finish time (EF) Latest finish time (LF) Latest start time (LS) Float time (m)
0 0 0 0 0 0
4 0 4 6 2 2
5 0 5 5 0 0
4 5 9 10 6 1
3 5 8 9 6 1
4 5 9 9 5 0
5 9 14 15 10 1
5 8 13 15 10 2
6 9 15 15 9 0
5 15 20 20 15 0
0 20 20 20 20 0
Critical path
I-2-5-8-9-E
C.-T. Chen, S.-F. Huang / Information Sciences 177 (2007) 2448–2458
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Table 7 Calculation result of X3 Task
I
1
2
3
4
5
6
7
8
9
E
Operation time Earliest start time (ES) Earliest finish time (EF) Latest finish time( LF) Latest start time (LS) Float time (m)
0 0 0 0 0 0
7 0 7 7 0 0
2 0 2 7 5 5
6 7 13 13 7 0
5 2 7 13 8 6
3 2 5 17 14 12
8 13 21 21 13 0
7 7 14 21 14 7
4 7 11 21 17 10
3 21 24 24 21 0
0 24 24 24 24 0
Critical path
I-1-3-6-9-E
Table 8 Critical paths with different a and k values a
k
Critical path
a¼0
k¼0 k ¼ 0:5 k¼1
a ¼ 0:5
k¼0 k ¼ 0:5 k¼1 –
I-2-5-8-9-E I-2-5-8-9-E I-2-5-8-9-E I-2-3-6-9-E I-2-5-8-9-E I-2-5-8-9-E I-2-5-8-9-E I-2-5-8-9-E
a¼1
this paper for solving the fuzzy PERT problems. Besides, a computer program has been developed to deal with the larger number of arguments (fuzzy numbers). Using this computer program, managers can determine the activity criticalities and find the critical path more easily and efficiently. 6. Conclusions PERT (program evaluation and review technique) is the most widely used technique for planning and coordinating large-scale projects. The main assumption in PERT is that the activity durations in a project can be estimated precisely and that they are statistically independent. As a result, in cases where this assumption does not hold, PERT appears to lead to poor estimation and inadequate management responses. In this study, FPERT methods were employed to overcome this problem. Triangular fuzzy numbers were used to represent the activity durations in a project network. Fuzzy bounds were computed for the starting and finishing times in each activity. A critical degree index based on the fuzzy float time was defined to calculate the critical degree of each activity and path. A second index was used to analyze the possibility of meeting a specified required time for a project network in an unpredictable environment. Using the proposed model, information on the status of the project network is immediately available and sensitivity analysis can be conducted by varying the durations of activities on the critical path. Acknowledgements The authors gratefully acknowledge the financial support of Taiwan’s National Science Council (Project number NSC 92-2213-E-212-008). The authors also express appreciation to Ms. C. Troy and Mr. Philip du Plessis for their editorial assistance. References [1] S. Avraham, Project segmentation – a tool for project management, International Journal of Project Management 15 (1997) 15–19. [2] A. Azaron, C. Perkgoz, M. Sakawa, A genetic algorithm approach for the time-cost trade-off in PERT networks, Applied Mathematics and Computation 168 (2005) 1317–1339.
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