Evaluating fuzzy earned value indices and estimates by applying alpha cuts

Expert Systems with Applications 38 (2011) 8193–8198

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

Evaluating fuzzy earned value indices and estimates by applying alpha cuts Leila Moslemi Naeni a, Amir Salehipour b,⇑ a b

School of Industrial Engineering, Sharif University of Technology, Tehran, Iran School of Industrial Engineering, Islamic Azad University-South Tehran Branch, Tehran, Iran

a r t i c l e

i n f o

a b s t r a c t

Keywords: Fuzzy earned value Alpha cut Uncertainty Project progress

The earned value technique is an essential technique in analyzing and controlling the performance of a project by providing a more accurate measurement of both project performance and project progress. This paper presents an approach to deal with fuzzy earned value indices. This includes developing new indices under fuzzy circumstances and evaluating them using alpha cut method. The model improves the applicability of the earned value techniques under real-life and uncertain conditions. A small example illustrates how the new model can be implemented in reality. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction

ics to make it more applicable and flexible. Lipke et al. (2009) provided a reliable forecasting method of completion cost and duration to improve the capability of project managers for making informed decisions. Recently Moslemi Naeni et al., (2010) have worked on fuzzy earned value and applied degree of possibility method to evaluate estimates. The motivation behind this paper is derived from the fact that despite the uncertain nature of the activities’ progress involved in a project, they are considered deterministic in all available EV techniques. In reality the activities’ data come from people’s judgments; hence they carry some degree of uncertainty. Bringing this uncertainty into interpretations, not only helps in measuring better performance and progress of a project, but also in extending the applicability of the EV techniques under the real-life and uncertain conditions. The major contribution of this paper is to develop an approach to deal with fuzzy earned value indices and estimates when measuring project performance and project progress. Through the paper, our terminology is based on the PMBOK guideline (PMI, 2004). For the simplicity, by ‘‘activity’’, we mean both activities and work packages. The remaining of this paper is organized as follows: Section 2 brings an introduction into the earned value and fuzzy theory. The section forms the basis of the proposed approach of Section 3. Evaluation and interpretation of fuzzy earned value indices and estimates are covered in Section 3. For clarification purposes, a simple example is studied in details in Section 4. The paper ends with the conclusion.

Earned Value Management (EVM/EV) is a project management technique developed to measure project progress in an objective manner. According to Project Management Institute (PMI),1 when properly applied, EVM provides an early warning of performance problems. The EV measures project performance and progress by efficiently integrating management of three most important elements of a project, i.e. cost, schedule and scope. In fact, it calculates cost and time performance indices of a project, estimates completion cost and completion time of a project, and measures project performance and project progress. Although being introduced in 2000 in PMBOKÒ guide (PMI, 2000), the first complete guide on the EV has been published in 2005 (PMI, 2005). Despite widely believed that implementing the EV techniques has many advantages and would enhance cost and schedule performances of a project; the research on the EV is very limited. Lipke (1999) developed cost and schedule ratio to manage cost and schedule reserves in projects. Later he introduced the earned schedule (ES) concept to outperform limitations of the historical EV schedule variance (SV) and schedule performance index (SPI) (Lipke, 2003). His studies were followed by Henderson (2003, 2004) and Vandevoorde and Vanhoucke (2005), where applicability and reliability of the ES were discussed. Anbari (2003) improved the effectiveness of EV implementation. Kim, Wells, and Duffey (2003) studied the implementation of the EV in different types of organizations and projects. Cioffi (2006) studied the EV mathemat-

2. The fuzzy earned value measurement technique ⇑ Corresponding author. E-mail addresses: [email protected] (L. Moslemi Naeni), asalehipour@ ymail.com (A. Salehipour). 1 In a more accurate definition by PMI (2005), the EVM combines measurements of technical performance (i.e. accomplishment of planned work), schedule performance (i.e. behind/ahead of schedule), and cost performance (i.e. under/over budget) within a single integrated methodology. 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.12.165

The earned value (EV) is a set of techniques to assist project managers in measuring and evaluating project progress and project performance by estimating completion cost and completion time of a project (based on its actual cost and actual time up to any given point in the project). The EV of an activity represents

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µ

Table 1 The EV measurement techniques. Activity product

Activity duration

Tangible

Fixed formula

Intangible

1 or 2 measurement periods

Very low

1

Less than half

Low

More than half

Half

Very high

High

More than 2 measurement periods

Weighted milestone percent complete Level of effort apportioned effort

0.1 the budgeted cost of work performed, and indicates how efficiently the project team utilizes the project resources. Table 1 presents a list of available techniques in calculating the EV of either an activity or a project. More details can be found in ‘‘practice standard for earned value management’’ (PMI, 2005). We explain here only the percent complete technique which forms the basis for fuzzy EV discussed here. In the percent complete, one of the simplest techniques for measuring the EV, in each measurement period the person-in-charge makes an estimate of the percentage of the activity completed, e.g. 26%. This technique can be the most subjective of the EV measurement techniques if there are no objective indicators based on which the estimates should be made. This greatly incorporates into errors and uncertainty which cause biased judgments. An idea to overcome this problem is to use the linguistic terms in estimating the completion percent of each activity, as the imprecise and uncertain data of activity performance and activity progress are common to arise. Fuzzy theory (Zadeh, 1965) explains uncertainty in events and systems where uncertainty arises due to vagueness or fuzziness rather than randomness alone. It is reasonable to model and treat the uncertainty using the linguistic terms with the fuzzy theory. For instance, if an activity progress cannot be stated in certainty, using linguistic terms it may be stated as ‘‘very low’’, etc.2 Clearly, this linguistic term cannot be applied on the EV technique before transforming it to a number. Thus, first we have to convert linguistic terms into fuzzy numbers by applying fuzzy principles. Typically, the project experts perform this transformation in accordance with their knowledge and their experience about the project and according to the activity attributes. Then we should modify the EV mathematics to consider fuzzy numbers. The application of the proposed method arises in situations where the total amount of work required to accomplish the activities is unknown or uncertain, and is out of control. Examples are, in a dam construction project the ground should be excavated until hard layer of rock is reached. Before reaching this layer, the exact amount of the operations and the required work are unknown, and also this is out of our control, so the percent complete of excavation activity cannot exactly be measured. In medical research projects and drug development projects, a majority of resources are devoted to the clinical experiments aims at testing the new drug for its benefits and potential side effects. The exact amount of work required to derive scientific conclusions is unknown in advance. In these cases and many other similar cases, it would be better and easier to evaluate the percentage of the activity completed by linguistic terms rather to evaluate it exactly and deterministically. We strongly believe the developed technique reflects better the uncertain nature of a project. The example below clarifies the core idea of this paper. Assume the completion percent of an activity includes uncertainty and is expressed as ‘‘half’’. As mentioned earlier the project expert transforms this into a fuzzy number by assigning a membership function3 to this linguistic term (like the one showed in Fig. 1. In 2 Other linguistic terms include but not limited to less than half, half, more than half, etc. 3 e quantifies the grade of membership of the element The membership degree l AðxÞ e x to the fuzzy set A.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Progress

Fig. 1. A fuzzy membership including trapezoidal and triangular fuzzy numbers and the corresponding linguistic terms.

Table 2 The assigned fuzzy numbers to each linguistic term of Fig. 1. Linguistic term

Fuzzy number

Very low Low Less than half Half More than half High Very high

[0,0, 0.1, 0.2] [0.1, 0.2, 0.2, 0.3] [0.2, 0.3, 0.4, 0.5] [0.4, 0.5, 0.5, 0.6] [0.5, 0.6, 0.7, 0.8] [0.7, 0.8, 0.8, 0.9] [0.8, 0.9, 1, 1]

figure the horizontal axis refers to the progress and is on a scale of 1). The summary of the transformation associated with Fig. 1 is shown in Table 2. Obviously, the output of this transformation is a fuzzy number. Note that Fig. 1 and Table 2 are only an example.4 For instance, according to Fig. 1 and Table 2, the linguistic term ‘‘half’’ equals to the fuzzy number [0.4, 0.5, 0.5, 0.6]. In general, the membership function of a trapezoidal fuzzy number, for example ~ ¼ ½a; b; c; d is defined as below: A

8 0; > > > > > > < ðx  aÞ=ðb  aÞ; lA~ ðxÞ 1; > > > > ðx  cÞ=ðd  cÞ; > > : 0;

x
ð1Þ

cd

If b = c a trapezoidal fuzzy number becomes triangular. Furthermore, a triangular fuzzy number can also be represented as a trapezoidal fuzzy number [a, b, b, d] or [a, c, c, d], for the sake of simplicity. This enables the authors to explain much easier their novel fuzzy-EV approach. We implemented the trapezoidal and triangular fuzzy numbers as these are the simplest fuzzy numbers. Thus their mathematics can be derived easily. f i , should be derived (Eq. (2)) At first, the fuzzy EV of activity i, EV which itself is based on fuzzy completion percent of the activity i (Eq. (3)).

fi ¼ e EV F i  BAC i ¼ ½E1i ; E2i ; E3i ; E4i 

ð2Þ

where

e F i ¼ ½a1i ; a2i ; a3i ; a4i 

ð3Þ

The BACi is the budget at completion of activity i and denotes the planned budget to complete the activity i. To derive the total fuzzy EV in each measurement period, someone should sum up f i for i = 1, . . . , n (n is the total number of project activities): all EV

4 Completion percent of an activity can be expressed using terms ‘‘approximately x’’ or ‘‘between x and y’’. This way is more suitable when dealing with long duration activities.

L. Moslemi Naeni, A. Salehipour / Expert Systems with Applications 38 (2011) 8193–8198

f ¼ EV

n X i¼1

" fi ¼ EV

n X

E1i ;

i¼1

n X

E2i ;

i¼1

n X

E3i ;

n X

i¼1

8195

# E4i ¼ ½E1 ; E2 ; E3 ; E4 

i¼1

ð4Þ f i can also be deterministic Note that in this equation some EV numbers rather than fuzzy numbers. Eqs. (1)–(4) that show how the percent complete of an activity can be expressed as a fuzzy number form the basis of developing fuzzy EV indices which are discussed below. 2.1. Fuzzy performance indices Schedule performance index (SPI) and Cost performance index (CPI) are two commonly used indices in the EV technique. SPI is a measure of the conformance of actual progress to schedule while CPI is a measure of the budgetary conformance of the actual cost of f (fuzzy SPI), and CPI f (fuzzy work performed. Before developing SPI CPI) let us bring the basic operations of fuzzy numbers. Assume ~ and n ~ are two trapezoidal fuzzy numr P 0 is a real number and m ~ ¼ ½m1 ; m2 ; m3 ; m4  and bers each with four members, i.e. m ~ ¼ ½n1 ; n2 ; n3 ; n4 . The basic operations of these two fuzzy numbers n are defined as follows.

~ þn ~ ¼ ½m1 þ n1 ; m2 þ n2 ; m3 þ n3 ; m4 þ n4  m ~ ~ m  n ¼ ½m1  n4 ; m2  n3 ; m3  n2 ; m4  n1  ~ n ~ ¼ ½m1  n1 ; m2  n2 ; m3  n3 ; m4  n4  m ~ n ~ ¼ ½m1 =n4 ; m2 =n3 ; m3 =n2 ; m4 =n1  m= ~  r ¼ ½m1  r; m2  r; m3  r; m4  r m

ðiÞ ðiiÞ ðiiiÞ ðivÞ ðvÞ

As SPI is the ratio of EV to the planned value (PV), i.e. SPI = EV/PV, f is derived using Eq. (5) (since EV is a fuzzy number; hence SPI is SPI a fuzzy number too)

f ¼ EV f =PV ¼ ½E1 =PV; E2 =PV; E3 =PV; E4 =PV;  SPI

ð5Þ

where PV known also as the budgeted cost of work scheduled (BCWS) is planned to be used during the project. The CPI is the most useful index indicating the cumulative cost efficiency of a project, thus it is the ratio of EV to the actual cost f is: (AC) or CPI = EV/AC. The CPI

f ¼ EV f =AC ¼ ½E1 =AC; E2 =AC; E3 =AC; E4 =AC CPI

ð6Þ

where AC known also as the actual cost of work performed (ACWP), indicates the resources used to achieve the actual work performed. Also the critical ratio (CR), called as schedule cost index (SCI) could easily be developed as a fuzzy number. CR is a multiplication of SPI by CPI and shows the project health5:

f ¼ SCI f ¼ SPI f  CPI f CR ¼ ½E21 =PV  AC; E22 =PV  AC; E23 =PV  AC; E24 =PV  AC

ð7Þ

2.2. Fuzzy estimates at completion Cost and time estimates at completion are important estimates with respect to project progress evaluation. A common formula to calculate cost estimate at completion (EAC) assumes the future trend of the project cost performance remains intact. In this formula EAC is calculated by dividing the Budget at Completion (BAC) by CPI. Thus, it assumes that CPI would be fixed during the rest of the project. The fuzzy EAC is:

5

For instance, CR = 1 implies that the overall project performance is on target.

Fig. 2. The ES versus the EV.

BAC g ¼ BAC ¼  EAC  E1 E2 E3 E4 f ; ; ; CPI AC AC AC AC   BAC  AC BAC  AC BAC  AC BAC  AC ; ; ; ¼ E4 E3 E2 E1

ð8Þ

EAC could also be influenced both by current cost performance and current schedule performance indices. This assumption forms another formula to derive EAC (Eq. (9)).

h i g ¼ AC þ ðBAC  EV f f Þ= SCI EAC   BAC  E4 BAC  E3 BAC  E2 BAC  E1 ¼ AC þ ;AC þ ;AC þ ;AC þ SCI4 SCI3 SCI2 SCI1 ð9Þ PMI (2005) reports several other formulas to derive EAC based on the EV data. Obviously, their fuzzification could be established quite similarly. Despite being developed to manage both time and cost, the majority of the research behind the EV technique has been through the cost aspects (Fleming & Koppelman, 2005). Hence the EV technique has not been widely used to estimate the total time at completion (total project duration). Nevertheless, there are three different techniques in estimating the completion time among which the third technique developed by Lipke (2003) and extended by Henderson (2004) is the most reliable and the most practical one (Vandevoorde & Vanhoucke, 2005).  The planned value (PV) technique of Anbari (2003).  The earned duration (ED) technique of Jacob (2003) and Jacob and Kane (2004).  The earned schedule (ES) technique of Lipke (2003). In the first two techniques, SPI does not reflect the real performance of the project as it tends to be 1 at the end of the project. Here, in developing the fuzzy time estimate at completion we benefitted from the ES technique. To avoid confusion in notation, we follow EACt to reflect the time estimate at completion (project duration). Earned schedule (ES) concept first introduced by Lipke (2003) as the time equivalent of the EV. According to Fig. 2, ES is resulted by projecting the EV on the baseline. Thus it measures schedule performance based on time. Eq. (10) illustrates mathematics of ES.

ES ¼ N þ ððEV  PV N Þ=ðPV Nþ1  PV N ÞÞ

ð10Þ

In Eq. (10) N is the longest time interval in which PVN is less than EV, PVN is the planned value at time N, and PVN+1 is the planned value at the next time interval, i.e. time N + 1 (Vandevoorde & Vanhoucke, 2005). Someone can find that Eq. (10) is an interpolation formula.

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The fuzzy ES could easily be established by Eq. (11). Note that each of the four members of the trapezoidal fuzzy number EV is projected on the baseline and forms a member of the trapezoidal fuzzy number ES.

ESi ¼ Ni þ ððEi  PV N Þ=ðPV Nþ1  PV N ÞÞ;

i ¼ 1; 2; 3; 4

¼ ½ES1 ; ES2 ; ES3 ; ES4 

f ES ð11Þ

In contrast to the SPI, SPIt is expressed in time units and is the ratio of ES to the actual duration (AD), i.e. SPIt = ES/AD. The fuzzification of SPIt is:

f f 1 =AD; ES f 2 =AD; ES f 3 =AD; ES f 4 =AD f t ¼ ES=AD ¼ ½ ES SPI

ð12Þ

The generic ES-based equation to estimate the time at the completion of a project is:

EAC t ¼ AD þ ððPD  ESÞ=PFÞ

ð13Þ

where PD is the planned duration to complete the project and PF is the performance factor which depends on the project status. The following three cases are of interest for PF:  PF = 1: The duration of the remained activities is as planned.

g t ¼ AD þ ðPD  ESÞ f EAC ¼ ½AD þ PD  ES4 ; AD þ PD  ES3 ; AD þ PD  ES2 ; AD þ PD  ES1 

ð14Þ

f t : The duration of remained activities changes with the  PF ¼ SPI f t trend. current SPI   g t ¼ AD þ ðPD  ES= f SPI f tÞ EAC   PD  ES4 PD  ES3 PD  ES2 PD  ES1 ¼ AD þ ; AD þ ;AD þ ;AD þ SPIt4 SPIt3 SPIt2 SPIt1 ð15Þ f It : The duration of remained activities changes with SC f It  PE ¼ SC f It ¼ SPI f t  CPI. f trend, where SC   g t ¼ AD þ ðPD  ES= f SC f It Þ EAC   PD  ES4 PD  ES3 PD  ES2 PD  ES1 ; AD þ ; AD þ ; AD þ ¼ AD þ SCIt4 SCIt3 SCIt2 SCIt1 ð16Þ

3. Interpretation of fuzzy performance indices and estimates The developed fuzzy-based EV indices and estimates should be interpreted to have an inference regarding the project progress and its status. Similar to the traditional EV indices,6 the developed fuzzy indices must be compared against value 1. As the indices are fuzzy numbers, this comparison should be made according to the available methods. Out of different methods proposed to compare the fuzzy numbers in the literature (see Bortolan & Degani, 1985; Mabuchi, 1988), a well-known fuzzy ranking method called a-cut introduced by Adamo (1980) is implemented in this paper. In the a-cut method, a particular value for a where 0 6 a 6 1 is e and B e are derived as chosen, then a-cuts for two fuzzy numbers A e e a A ¼ ½a; b and a B ¼ ½c; d, respectively. Eq. (17) is a general statee against B e with a-cut method (see also Fig. 3): ment comparing A

e6B e ¼ b 6 aB e ¼d e if a A A R R

~ and B. ~ Fig. 3. a-Cut for fuzzy numbers A

ð17Þ

6 In the EV indices, if SPI < 1, SPI > 1, and SPI = 1, then the project is behind the schedule (it will finish late), the project is ahead of the schedule (it will finish sooner than expected) and the project is on the schedule (the project will finish according to the schedule,), respectively. The same interpretation applies to the CPI except that the CPI deals with the cost, hence if CPI < 1 (>1 or =1) then the project is over (under or within) the budget, respectively.

e is less than the fuzzy According to Fig. 3, the fuzzy number A e is less than the e number B if the right-hand-side of a-cut of A e In brief, comparison using aright-hand-side of the a-cut of B. cut method means the relative relationship of two fuzzy numbers can be determined by comparing their endpoint of the close intervals generated by the a-cut. The a-cut method leads to a deterministic conclusion. In practice, the choice of appropriate value for a can be based on the decision maker and on the past experience, and/or on the case under study. Normally, we can use the past estimates and their actual values to derive a, i.e. choosing a such that the gap between cost and time estimates at completion and their actual values of the past similar projects is minimized. The sensitivity analysis technique can be implemented to perform such an analysis. Also the case under study may affect the choice of a. An example could be a new product development project. Because of a high degree of risk associated, it might be beneficial to consider members with small membership degrees (cases with small possibility of occurrence), thus a is preferred to be small.7 3.1. Analysis of fuzzy cost and time indices To evaluate the fuzzy EV indices of Section 2, first the a-cuts should be derived. We know for the trapezoidal fuzzy number e ¼ ½a; b; c; d, its a-cut can be obtained using Eq. (18). A ae

A ¼ ½a þ aðb  aÞ; d  aðd  cÞ;

06a61

ð18Þ

f (Eq. (5)) and CPI f (Eq. Now the right-hand-side of a-cut of SPI a f (6)) indices, i.e. and R CPI are calculated as Eqs. (19) and (20) illusf and the trates. Also Tables 3 and 4 interpret the comparison of aR SPI a CPI f against value 1, respectively. R af

R SPI

¼ ðE4 =PVÞ  aððE4  E3 Þ=PVÞ ¼ ðE4  aðE4  E3 ÞÞ=PV

ð19Þ

af

¼ ðE4 =ACÞ  aððE4  E3 Þ=ACÞ ¼ ðE4  aðE4  E3 ÞÞ=AC

ð20Þ

R CPI

3.2. Analysis of fuzzy cost and time estimates Usually, the budget at completion (BAC) which refers to the total assigned budget to complete the project is determined from the beginning of the project. The project therefore is expected to complete within this budget. Besides the BAC, there is the funding reserve (FR) budget to be used in unexpected circumstances. Thus, the FR provides risk mitigation of meeting the planned cost and decreases the probability of the budget overrun. Therefore, the total available budget (TAB) is summation of BAC and FR. In the traditional models, in each evaluation period the completion cost of the project is estimated as EAC and compared with BAC and TAB. Despite the method being used when estimating the required cost at completion, the questions are how the EAC should be 7 In general choosing a close to 1 means only members with a large membership degree are considered in the comparison.

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L. Moslemi Naeni, A. Salehipour / Expert Systems with Applications 38 (2011) 8193–8198 Table 3 Interpretation of fuzzy SPI.

4. Example

Status

Interpretation

a SPI f <1 R

Behind schedule

a SPI f ¼1 R

Within schedule

a SPI f >1 R

Ahead of schedule

In this section we bring a small example illustrating the developed approach. The example refers to a small work package consisting of four activities over a period of 12 months. Table 7 shows the planned values (PV) and actual costs (AC) up to month 4. The data regarding activity progress and activity budget at completion (BAC) are brought in Table 8. Assume here we would like to answer the following questions. f f and ES?  What are the total EV f SPI f and SPI f t?  What are the status of CPI;  What are the cost and time estimates at completion of this work package?

Table 4 Interpretation of fuzzy CPI. Status

Interpretation

a CPI f <1 R

Over budget

a CPI f ¼1 R

Within budget

a CPI f >1 R

Under budget

interpreted against the BAC, and whether the TAB can cover the project until its completion. Assume the cost at completion which has been determined in a g ¼ ½a; b; c; d. After finding a EAC g , it must be comgiven period is EAC R pared to the fixed BAC and TAB. Table 5 interprets this comparison. In the EV techniques the total available time for the project is called planned duration (PD) and usually is decided at the beginning of the project. The negotiated completion date (NCD) is the final date by when the project must complete. The difference of NCD and PD is called schedule reserve (SR), and could be of help under unexpected circumstances. In fact, SR decreases the probability of overdue, and provides risk mitigation of meeting the planned duration. As mentioned earlier, using the ES technique to derive EACt yields more reliable results. Using the right-hand-side value of ag t against the PD and NCD are interpreted cut to compare the EAC in Table 6.

Linguistic terms stating the progress of activities 1, 2 and 3 (Table 8) are transformed into fuzzy numbers ( e F i ) using Table 1 (see Table 9 for the result of transformation). According to the BAC of each activity, the fuzzy EV of each activity calculated using Eq. (2) is

f1 ¼ e EV F 1  BAC 1 ¼ ½700; 800; 800; 900 Applying Eq. (2) for activities 2 and 3 will result in:

f 2 ¼ ½40; 80; 160; 200 and EV

f 3 ¼ ½0; 0; 120; 240 EV

Using Eq. (4) we can find the total EV of the work package, i.e. the total EV for all of the four activities:

f ¼ EV

4 X

f i ¼ ½1040; 1180; 1380; 1640 EV

i¼1

As Table 7 illustrates, the total planned value and the total actual cost for the work package at the data date, i.e. month 4 are 1200$ and 1500$, respectively. Considering a = 0.8, the work package performance in comparison with SPI and CPI is interpreted as follows.

 f  f ¼ EV ¼ 1040 ; 1180 ; 1380 ; 1640 ¼ ½0:69; 0:79; 0:92; 1:09 CPI 1500 1500 1500 1500 AC f ¼ 1:09  0:8ð1:09  0:92Þ ¼ 0:95 thus aR CPI

Table 5 g , BAC and TAB. Interpreting the comparison among aR EAC Status

 f  f ¼ EV ¼ 1040 ; 1180 ; 1380 ; 1640 ¼ ½0:87; 0:98; 1:15; 1:37 SPI PV 1200 1200 1200 1200

Interpretation

a EAC g 6 BAC R

Project will complete under the BAC

g 6 TAB BAC < aR EAC a EAC g > TAB

Project will complete but uses the FR

f ¼ 1:37  0:8ð1:37  1:15Þ ¼ 1:19 thus aR SPI f is less than 1 and the a SPI f is greater than 1, thus this The aR CPI R work package is behind the planned budget and ahead of the planned schedule.

Project can not complete by using the TAB

R

Table 6 g t , PD and NCD. Interpreting the comparison among aR EAC

Table 8 The data of work package of example.

Status

Interpretation

Activity

BAC ($)

Progress

a EAC g 6 PD R

Project will finish before PD

g t 6 NCD PD < aR EAC a EAC g t > NCD

Project needs the SR to finish

1 2 3 4

1000 800 1200 2000

High Between 10% and 20% Very low 15%

Project can not finish by the NCD

R

Table 7 PV and AC of example. Month

1

2

3

4

5

6

7

8

9

10

11

12

PV AC

100 200

400 500

750 1000

1200 1500

1700 –

2300 –

2950 –

3600 –

4150 –

4650 –

4850 –

5000 –

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Table 9 The percent complete and the EV of activities. Activity

BAC ($)

Progress

e Fi

fi EV

1 2

1000 800

[0.7, 0.8, 0.8, 0.9] [0.05, 0.1, 0.2, 0.25]

[700, 800, 800, 900] [40, 80, 160, 200]

3 4

1200 2000

High Between 10% and 20% Very low 15%

[0, 0, 0.1, 0.2] –

[0, 0, 120, 240] 300

5. Conclusion

Table 10 g t calculations. The summary of EAC E PV N

i

Ei

Ni

Ei  PV Ni

PV Ni þ1  PV Ni

ESi ¼ N i þ PV Ni þ1 PVi N

1 2 3 4

1040 1180 1380 1640

3 3 4 4

290 430 180 440

450 450 500 500

3.64 3.96 4.36 4.88

i

i

Now, we would like to estimate the completion cost of the work package. Assuming that the trend affecting CPI is kept fixed during the rest of the project, the completion cost of the work package is estimated as follows.

  g ¼ BAC ¼ BAC  AC ; BAC  AC ; BAC  AC ; BAC  AC EAC f E4 E3 E2 E1 CPI ¼ ½4573; 5435; 6356; 7212 By Table 8, the total BAC for all activities is 5000$. Assuming FR for this work package is 500$, the cost estimate at completion would be interpreted as follows.

TAB ¼ BAC þ FR ¼ 5000 þ 500 ¼ 5500 a

f Having the above calculations we can derive 0:8 R SPI and 0:8 g ¼ 12:34. R EAC t > PD implies that if the operations on this work package continue with this performance into the rest of the project, project duration should be increased.

0:8 g R EAC t

g ¼ 7212  0:8ð7212  6356Þ ¼ 6527

R EAC

a EAC g > TAB implies that if the operations on this work package R continue with this performance into the rest of the project, more budget will be required. Usually, under this condition, the project manager should implement the corrective actions to reduce the costs. f each of the four members of the trapezoidal EV f To calculate ES, is projected on the baseline and forms members of the trapezoidal fuzzy number ES (Eq. (11)). For instance, the calculations behind ES1 are:

N 1 ¼ 3 as PV 3 < E1 ¼ 1040 < PV 4 thus   E1  PV 3 1040  750 ¼3þ ¼ 3:64 ES1 ¼ N1 þ PV 4  PV 3 1200  750 f ¼ ½3:64; 3:96; 4:36; 4:88 as Table 10 Following these calculations, ES shows. g t (Eq. (15) given PF ¼ SP f t (Eq. (12)) and EAC f t ) we To derive SPI have

f f t ¼ ES=AD ¼ ½3:64=4; 3:96=4; 4:36=4; 4:88=4 SPI ¼ ½0:91; 0:99; 1:09; 1:22   g t ¼ 4 þ 12  4:88 ; 4 þ 12  4:36 ;4 þ 12  3:96 ;4 þ 12  3:64 EAC 1:22 1:09 0:99 0:91 ¼ ½9:84;11:01; 12:13;13:17

The percent complete is the simplest and the most applied technique to measure the earned value; however it has the disadvantage of using subjective judgments when describing the percent of completed work. On the other hand, the fuzzy estimates of both completion cost and time can assist project managers to estimate the future status of the project in a more applicable way. Additionally, there are situations that the developed approach is more practical than the traditional earned value. In this paper, we used the linguistic terms and fuzzy theory to measure the earned value and evaluate its associated indices and estimates. In the evaluation and interpretation of the fuzzy indices and estimates we benefited from the a-cut method. As future research directions, we are working on how a more reliable method to track those indices and estimates over time could be established. Furthermore, we are studying other fuzzy number ranking methods to interpret the developed indices and estimates. References Adamo, J. M. (1980). Fuzzy decision trees. Fuzzy Sets and Systems, 4, 207–219. Anbari, F. (2003). Earned value project management method and extensions. Project Management Journal, 34, 12–23. Bortolan, G., & Degani, R. (1985). A review of some methods for ranking fuzzy subsets. Fuzzy Sets and Systems, 15, 1–19. Cioffi, D. F. (2006). Designing project management: A scientific notation and an improved formalism for earned value calculations. International Journal of Project Management, 24, 136–144. Fleming, Q. W., & Koppelman, J. M. (2005). Earned value project management (3rd ed.). Project Management Institute. Henderson, K. (2003). Earned schedule: A breakthrough extensions to earned value theory? A retrospective analysis of real project data. The Measurable News(Summer), 13–23. Henderson, K. (2004). Further developments in earned schedule. The Measurable News(Spring), 15–22. Jacob, D. S. (2003). Forecasting project schedule completion with earned value metrics. The Measurable News(Spring), 7–9. Jacob, D. S., & Kane, M. (2004). Forecasting schedule completion using earned value metrics revisited. The Measurable News(Summer), 11–17. Kim, E., Wells, W. G., Jr., & Duffey, M. R. (2003). A model for effective implementation of Earned Value Management methodology. International Journal of Project Management, 21, 375–382. Lipke, W. (1999). Applying management reserve to software project management. Journal of Defense Software Engineering, 17–21. Lipke, W. (2003). Schedule is different. The Measurable News(Summer), 31–34. Lipke, W., Zwikael, O., Henderson, K., & Anbari, F. (2009). Prediction of project outcome, the application of statistical methods to earned value management and earned schedule performance indexes. International Journal of Project Management, 27, 400–407. Mabuchi, S. (1988). An approach to the comparison of fuzzy subsets with a-cut dependent index. IEEE Transactions on Systems, Man, and Cybernetics, 18, 264–272. Moslemi Naeni, L., Shadrokh Sh., and Salehipour, A. (2010). A fuzzy approach for the earned value management. International Journal of Project Management. To appear. PMI (2000). Project management body of knowledge (PMBOK) (2nd ed.). Project Management Institute. PMI (2004). Project management body of knowledge (PMBOK) (3rd ed.). Project Management Institute. PMI (2005). Practice standard for earned value management. Project Management Institute. Vandevoorde, S., & Vanhoucke, M. (2005). A comparison of different project duration forecasting methods using earned value metrics. International Journal of Project Management, 24, 289–302. Zadeh, L. (1965). Fuzzy sets. Information and Control, 8, 338–353.