A fuzzy approach for the earned value management

International Journal of Project Management 29 (2011) 764 – 772 www.elsevier.com/locate/ijproman

A fuzzy approach for the earned value management Leila Moslemi Naeni a , Shahram Shadrokh a , Amir Salehipour b,⁎ a

b

Department of Industrial Engineering, Sharif Univ. of Tech., Tehran, Iran Department of Industrial Engineering, School of Engineering, Tarbiat Modares Univ., Tehran, Iran Received 11 August 2008; received in revised form 20 June 2010; accepted 27 July 2010

Abstract The earned value technique is a crucial technique in analyzing and controlling the performance of a project which allows a more accurate measurement of both the performance and the progress of a project. This paper presents a new fuzzy-based earned value model with the advantage of developing and analyzing the earned value indices, and the time and the cost estimates at completion under uncertainty. As the uncertainty is inherent in real-life activities, the developed model is very useful in evaluating the progress of a project where uncertainty arises. A small example illustrates how the new model can be implemented in reality. © 2010 Elsevier Ltd. and IPMA. All rights reserved. Keywords: Fuzzy earned value; Earned value technique; Uncertainty; Project progress

1. Introduction Earned Value Management (EVM) is a project management technique used 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. Referring EVM to EV (Earned Value) most often, the EV measures the project performance and the project progress by integrating efficiently the management of the three most important elements in a project, i.e. cost, schedule and scope. In fact, it calculates the cost and time performance indices of a project, estimates the completion cost and the completion time of a project, and measures the performance and the progress of a project by comparing the planned value and the actual costs of activities to their corresponding earned values. ⁎ Corresponding author. E-mail addresses: [email protected] (L.M. Naeni), [email protected] (S. Shadrokh), [email protected] (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.

Although being introduced in 2000 in PMBOK® guide (PMI, 2000), the first complete guide on the EV has been published in 2005 (PMI, 2005). Nowadays, it is widely believed that implementing EV techniques has many advantages and would enhance the cost and the schedule performances of a project, but on the contrary the research on the EV is very limited. Lipke (1999) developed cost ratio and schedule ratio to manage the cost and the schedule reserves in projects. Later he introduced the Earned Schedule (ES) concept to resolve the limitations and peculiarities inherent in the use of the historical EV Schedule Variance (SV) and Schedule Performance Index (SPI) (Lipke, 2003). Henderson (2003 and 2004) studied the applicability and reliability of the ES. Also Vandevoorde and Vanhoucke (2005) concluded that the best and the most reliable method to estimate time at completion is the ES method. Anbari (2003) enhanced the effectiveness of EV implementation. Kim et al. (2003) studied the implementation of the EV in different types of organizations and projects. Lipke (2004) developed project cost and time performance probabilities. A new notation for the EV analysis is presented in Cioffi (2006) to make EV mathematics more transparent and flexible. Lipke et al. (2009) provided a reliable forecasting method of the final cost and duration to improve the capability of project managers for making informed decisions.

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L.M. Naeni et al. / International Journal of Project Management 29 (2011) 764–772

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 data regarding the activities come from people's judgments; hence they carry some degree of uncertainty. Considering this uncertainty into interpretations and calculations, not only helps in measuring better the performance and the 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 a new fuzzy-based EV technique to measure and evaluate the performance and the progress of a project and its activities under uncertainty. The developed model benefits from the linguistic terms and the Fuzzy theory. Through the paper, our terminology is based on the PMBOK guideline (PMI, 2004). For the simplicity, by “activity”, we mean both activities and the work packages. The remaining of this paper is organized as follows. Section 2 brings an introduction into the EV and its techniques. Section 3 explains briefly the fuzzy theory and its application to the EV. Developing the new fuzzybased EV technique and its interpretations are covered in Section 4. For clarification purposes, a simple example is studied in details in Section 5. The paper ends with the conclusion. 2. The earned value measurement techniques The EV consists of techniques to assist project managers in measuring and evaluating the progress and the performance of a project by estimating the 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 is a measure of the completed work and represents 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). To keep the paper short, we avoid explaining these techniques here, except the Percent Complete technique which is the basis of the calculation in the proposed model. In Percent Complete, one of the simplest techniques for measuring the EV, in each measurement period the person-incharge makes an estimate of the percentage of the activity complete, 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

Table 1 The EV measurement techniques. Activity product

Activity duration

Tangible

Fixed formula

Intangible

1 or 2 measurement periods

765

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. This forms the basis of our novel idea. 3. Using fuzzy theory to measure the earned value First introduced by Lotfi Zadeh (1965) fuzzy theory explains uncertainty in events and systems where uncertainty arises due to vagueness or fuzziness rather than due to 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 apply the fuzzy principles on the linguistic terms to convert them into fuzzy numbers. Then we modify the EV mathematics to reflect the new values (fuzzy numbers). Typically the project experts perform this transformation in accordance with their knowledge and their experience about the project, and with activity attributes. The application of the proposed method arises in situations where the total amount of the work required to perform the activities are unknown or uncertain, and is out of control. Examples are, in construction project of a dam the ground should be excavated until the 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 to include this uncertainty in our results, it would be better and easier to evaluate the percentage of the activity completed by linguistic terms rather to evaluate it exactly and deterministically. In fact, a linguistic term helps to estimate the activity progress easier by answering the question “What fraction/ percent of the activity is complete?” We strongly believe the new developed technique reflects better the uncertain nature of the 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 should transform this into a fuzzy number by assigning a membership function3 to this linguistic term, to express the relationship between the linguistic term and the

More than 2 measurement periods

Weighted milestone percent complete Level of effort Apportioned effort

2 Other linguistic terms include but not limited to less than half, half, more than half, high, very high, etc. 3 The membership degree μ Ã x quantifies the grade of membership of the element x to the fuzzy set Ã.

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L.M. Naeni et al. / International Journal of Project Management 29 (2011) 764–772

µ Very low

1

Less than half Low

0.1

0.2

More than half Half

0.3

0.4

0.5

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

Very high High

0.6

0.7

0.8

0.9

1 Progress

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]

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

fuzzy number (like the one showed in the Fig. 1. In this 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 examples.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 à = [a, b, c, d] is defined as below: 0 x−a

μà x =

b−a

1 x−c d−c

0

xba abxbb bbxb c cbxb d

1

xNd

Note that a trapezoidal fuzzy number becomes triangular when b = c. For the ease of representation and calculation, a triangular fuzzy number is also represented as a trapezoidal fuzzy number [a,b,b,d] or [a,c,c,d]. The trapezoidal fuzzy number and the triangular fuzzy number are both the simplest fuzzy numbers which their corresponding calculations can be performed easily. This enables the authors to explain much easier their novel Fuzzy-EV approach. In expressing the completion percent of each activity on the ˜ i can be basis of a fuzzy number, the fuzzy EV of activity i, EV calculated using Eq. (2). ˜i = F˜ i × BACi = ½E1i E2i E3i E4i EV

2

Where F̃i is the fuzzy completion percent of the activity i, i.e. F˜ i = ½a1i a2i a3i a4i

3

and the BACi is the Budget at Completion of activity i which denotes the planned budget to complete the activity i. The total ˜ , is calculated by fuzzy EV in each measurement period, EV

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. 4

˜ i for i = 1, …, n where n is the total summing up all the EV number of the project activities:  ˜ = ∑ni=1 EV ˜i = ∑ni= 1 E1i ∑ni= 1 E2i ∑ni= 1 E3i ∑ni= 1 E4i EV 4 = ½E1 E2 E3 E4 ˜ i can be fuzzy numbers while some In Eq. (4), some of the EV of them can be deterministic numbers. With Eqs. (1) to (4), other fuzzy EV indices can be developed similarly and we will discuss them thoroughly in Section 4. To conclude this section we should add that the new fuzzybased EV technique would result in a more practical technique in comparison to the traditional EV technique by incorporating greatly into a more realistic implementation and interpretation of the EV indices and estimates where measuring and evaluating the activity progress under uncertainty can be dealt very effectively. 4. Fuzzy-based earned value indices and estimates In the previous section we showed how the percent complete of an activity can be expressed as a fuzzy number. As we have a fuzzy percent complete and a fuzzy activity EV, all related indices and estimates have to be expressed as fuzzy numbers. This section is devoted to the development of these indices and estimates. 4.1. Fuzzy performance indices Two widely used indices in the EV technique are the Schedule Performance Index (SPI) and the Cost Performance Index (CPI). In this subsection we shall develop the fuzzy SPI, ˜ , and the fuzzy CPI,CPI ˜ . SPI Before going ahead, let us bring the basic operations of fuzzy ˜ and ñ are two numbers. Assume r ≥ 0 is a real number and m trapezoidal fuzzy numbers each with four members, i.e. ˜ = ½m1 m2 m3 m4 and ñ = [n1, n2, n3, n4]. The basic operam tions of these two fuzzy numbers are defined as follows. ˜ + n˜ = ½m1 + n1 m2 + n2 m3 + n3 m4 + n4 m

i

˜ n˜ = ½m1 −n4 m2 −n3 m3 −n2 m4 −n1 m−

ii

˜ × n˜ = ½m1 × n1 m2 × n2 m3 × n3 m4 × n4 m

iii

L.M. Naeni et al. / International Journal of Project Management 29 (2011) 764–772

˜ = n˜ = ½m1 = n4 m2 = n3 m3 = n2 m4 = n1 m

iv

˜ × r = ½m1 × r m2 × r m3 × r m4 × r m

v

The SPI is a measure of the conformance of actual progress to schedule. In other words, SPI is the ratio of EV to the Planned Value (PV), i.e. SPI = EV/PV. In the proposed fuzzy EV technique, EV is a fuzzy number; hence SPI is a fuzzy number ˜. too. Eq. (5) shows SPI ˜ = SPI

˜ EV PV

=

h  E

E2

1



PV

E3



PV

E4



PV

5

PV

PV is also known as the Budgeted Cost of Work Scheduled (BCWS) and is planned to be used during the project. CPI is a measure of the budgetary conformance of the actual cost of work performed. In other words, CPI is the most useful index indicating the cumulative cost efficiency of a project. CPI is the ratio of EV to the Actual Cost (AC), i.e. CPI = EV/AC. ˜ is calculated as follows: CPI ˜ = CPI

˜ EV AC

=

h  E

E2

1



AC

E3

AC



E4 AC



6

AC

AC is also known as the Actual Cost of Work Performed (ACWP) and is an indication of the resources that have been used to achieve the actual work performed. The Critical Ratio (CR) is a multiplication of SPI by CPI and shows the project health.5 It is also called Schedule Cost Index (SCI) which is calculated using Eq. (7). ˜ = SPI ˜ × CPI ˜ ˜ = SCI CR h 2 E22 = E1 PV × AC

7 E32

PV × AC

E42 PV × AC

PV × AC

4.2. Fuzzy cost estimate at completion Among several formulas available to calculate Cost Estimate at Completion (EAC), a common formula assuming the future trend of the project cost performance remains intact, is to divide the Budget at Completion (BAC) by the CPI, i.e. EAC = BAC/ CPI. In other words, this formula assumes that CPI would be fixed during the rest of the project. As in the proposed method CPI is a fuzzy number, EAC becomes a fuzzy number too, thus:

In the other formula to calculate the EAC, EAC will be influenced both by the current cost performance and current schedule performance indices (Eq. (9)). ˜ = AC + EAC

h

E4 AC

8

BAC × AC BAC × AC BAC × AC BAC × AC = E4 E3 E2 E1

5

For instance, if CR = 1 then the overall project performance is on target.

˜ BAC− EV ˜ SCI

BAC−E4 BAC−E3 BAC−E2 BAC−E1 = AC + AC + AC + AC + SCI4 SCI3 SCI2 SCI1

9 There are some other formulas which can be used to derive the EAC based on the EV data (For instance see PMI, 2005). Easily, the other fuzzy-based formulas can be derived in a similar way. 4.3. Fuzzy time estimate at completion Although the EV technique has been designed to manage both time and cost, majority of the research has been focused on the cost aspects (Fleming and Koppelman, 2005), thus the EV technique has not been widely used to estimate the total time at completion or total project duration based on the actual performance up to a given point in the project. Nevertheless, there are three different techniques in estimating the completion time: The Planned Value (PV) technique (Anbari, 2003) The Earned Duration (ED) technique (Jacob, 2003; Jacob and Kane, 2004) The Earned Schedule (ES) technique (Lipke, 2003; Henderson, 2004)

According to Vandevoorde and Vanhoucke (2005), among these three techniques, the third technique developed by Lipke (2003) and later extended by Henderson (2004) is the most reliable and the most practical technique. The first two techniques use SPI. As SPI tends to be 1 at the end of the project and does not reflect the real performance of the project, these two techniques are not reliable. Thus we use the ES technique to develop the fuzzy-based time estimate at completion. Since EAC reflects the estimate of the completion cost, we follow the EACt notation with index t to reflect the time estimate at completion. According to Lipke (2003), the ES is the time equivalent of the EV which is resulted by projecting the EV on the baseline. Thus it measures schedule performance using time. Calculation of the ES is determined by comparing EV against PV (Eq. 10). Fig. 2 explains the ES and the EV graphically. 

BAC ˜ = BAC =  EAC E1 E2 E3 ˜ CPI AC AC AC

767

ES = N +

EV − PVN PVN + 1 − PVN

10

Where 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 and Vanhoucke, 2005). Obviously, Eq. (10) is an interpolation formula. As in the fuzzy-based EV, EV is a fuzzy number (Eq. 2), then the ES will be a fuzzy number too. Thus each of the four

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Money

•

PV

˜ t : The duration of remained activities changes with PF = SCI ˜ ˜ t is the multiplication of the SPI ˜ t by the the SCI t trend. SCI ˜ , i.e. SCI ˜ t = SPI ˜ t × CPI ˜ (Eq. (16)). CPI ˜ t = AD + EAC



= AD +

˜ PD − ES

˜ t SCI

PD−ES4 PD−ES3 PD−ES2 PD−ES1 AD + AD + AD + SCIt4 SCIt3 SCIt2 SCIt1

EV

16 4.4. Interpretation of fuzzy performance indices

N

N+1

Actual Time

ES Fig. 2. The ES versus the EV.

members of the trapezoidal fuzzy number EV is projected on the baseline and forms a member of the trapezoidal fuzzy number ES. The mathematical calculations for the ES are as follows:  11 ESi = Ni + Ei − PVN PVN + 1 − PVN

˜ = ½ES1 ES2 ES3 ES4 i = 1 2 3 4 ES In contrast to SPI, SPIt is expressed in time units and is the ratio of the ES to the actual duration (AD), i.e. SPIt = ES / AD. Eq. (12) represents this. ˜t= SPI

˜ ES

=

AD

ES 

ES2

1

AD



ES3 AD



ES4 AD

 AD

12

The generic ES-based equation to estimate the time at the completion of a project, or the project duration is:  EACt = AD +

PD − ES PF

13

Where the PD is the planned duration to complete the project and the PF is the performance factor which depends on the project status. Usually the following three cases are considered for PF: •

PF = 1: The duration of the remained activities is as planned (Eq. (14)). ˜ t = AD + PD − ES ˜ EAC

14 ˜ t : The duration of remained activities changes with PF = SPI ˜ t trend (Eq. 15). the current SPI ˜ t = AD + EAC



= AD +

˜ PD − ES

 T A˜ ≥ B˜ = supx ≥ y min μA˜ x μB˜ y

˜t SPI

PD−ES4 PD−ES3 PD−ES2 PD−ES1 AD + AD + AD + SPIt4 SPIt3 SPIt2 SPIt1

15

17

The proposed method leads to deterministic conclusions almost in all cases, however in several cases the results are not deterministic which might be of interest as it is much closer to the nature of project and the proposed model in this paper. ˜ (Eq. (5)) and CPI ˜ (Eq. (6)), To apply the Eq. (17) to SPI however the comparison is made against 1, thus  ˜ ≥1 = supx≥1 min μ ˜ x 1 = supx≥1 μ ˜ x T SPI 18 SPI SPI  ˜ ≥1 = supx≥1 min μ ˜ x 1 = supx≥1 μ ˜ x T CPI CPI CPI



= ½AD + PD−ES4 AD + PD−ES3 AD + PD−ES2 AD + PD−ES1

•

After developing the fuzzy-based EV indices, we should interpret them to have an inference regarding the project progress and its status. Similar to the traditional EV indices where a comparison is made against the value 16 which shows the EV against planned value and actual cost, the same story goes here, i.e. the new fuzzy indices must be compared against value 1. But as the new indices are fuzzy numbers, we should compare them using the methods proposed for comparing the fuzzy numbers. Out of different methods proposed to compare fuzzy numbers in the literature (see Bortolan and Degani, 1985; Dubois and Prade, 1980), a well-known fuzzy ranking method proposed by Dubois and Prade (1980) evaluates the degree of possibility by which a fuzzy number is either greater or less than another fuzzy number.  According to Dubois and Prade (1980), T A˜ is the degree of  possibility of fuzzy number Ã, thus T A˜ = μà x where x ∈ Ã. ˜ the degree of possibility that Given two fuzzy numbers à and B, A˜ ≥ B˜ is

19

 ˜ Assuming  trapezoidal fuzzy numbers, thus both T CPI ≥1 and ˜ ≥1 results in five scenarios as Tables 3 and 4 show, T SPI respectively. In these tables the vertical line depicts the position of

6 In the EV indices, if SPI b 1, SPI N 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 b 1 (N 1 or = 1) then the project is over (under or within) the budget, respectively.

L.M. Naeni et al. / International Journal of Project Management 29 (2011) 764–772

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Table 3 ˜ ≥1 and CPI ˜ ≤1. The degree of possibility of CPI ˜ No. of State of CPI Scenario against 1

˜ ≥1 Degree of possibility of CPI

˜ ≤1 Degree of possibility of CPI  ˜ ≤1 = supx≥1 μ ˜ x T CPI CPI

=1

Behind the budget

 ˜ ≤1 = supx≥1 μ ˜ x T CPI CPI

=1

Approximately behind the budget

On the budget

Graphical Description

1

db1

 ˜ ≥1 = supx≥1 μ ˜ x T CPI CPI

=0

2

cb1bd

 ˜ ≥1 = supx≥1 μ ˜ x T CPI CPI

=

3

bb1bc

 ˜ ≥1 = supx≥1 μ ˜ x T CPI CPI

=1

 ˜ ≤1 = supx≥1 μ ˜ x T CPI CPI

=1

4

ab1bb

 ˜ ≥1 = supx≥1 μ ˜ x T CPI CPI

=1

 ˜ ≤1 = supx≥1 μ ˜ x T CPI CPI

=

5

aN1

 ˜ ≥1 = supx≥1 μ ˜ x T CPI CPI

=1

 ˜ ≤1 = supx≥1 μ ˜ x T CPI CPI

=0

d−1 d−c

˜ and CPI ˜ to this value. value 1, as we would like to compare SPI ˜ ≥1 and According to the tables, the degree of possibility of CPI   ˜ ≥1, i.e. T CPI ˜ ≥1 , and T SPI ˜ ≥1 are between 0 and 1. The SPI  ˜ ≥1 to 1, for instance, shows the project is proximity of T CPI ahead of budget, and its proximity to 0 shows the project is behind the budget. As mentioned earlier, this comparison does not always give crisp decisions, which is more appropriate and consistent with the basis of the developed model. For scenarios 2 and 4 in   Tables 3 and 4, the values of ˜ ≥1 and T SPI ˜ ≥1 are depicted in the graphical T CPI description by black bold lines (as a part of original red line). Thus, for instance in scenario 2 of Table 3, the length of the black bold line inside the trapezoidal is equal to  ˜ ≥1 = supx≥1 μ ˜ x T CPI CPI

=

d−1 d−c

Note that according to the fuzzy principles we  cannot ˜ ≤1 based on T CPI ˜ ≥1 , thus we draw conclusions for T CPI

1−a b−a

Decision Making

Approximately ahead of the budget

Ahead of the budget

 ˜ ≥1 separately (see Table 3). This rule calculated T CPI  ˜ ≤1 (Table 4). applies also to T SPI 5. Example In this section we bring a small example illustrating the basis of the new fuzzy-based EV calculations. This example is drawn from a recent case experienced by one of the authors and refers to a medical research project. Among the elements of level 1 of the WBS, existed the ‘Sample Finding’, that was divided into four work packages ‘University Sampling’, ‘School Sampling’, ‘Hospital Sampling’, and ‘Public Media Sampling’. Each of these work packages consisted of the six activities ‘Filling Questionnaire’, ‘Checking-up people’, ‘Getting sample’, ‘Testing sample’, ‘Analyzing sample’, and ‘Refining sample’. To keep the example simple enough to be understood easily by the readers, and as each of the above four work packages have similar activities, our example deals only with

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Table 4 ˜ ≥1 and SPI ˜ ≤1. The degree of possibility of SPI No. of Scenario

˜ State of SPI against 1

˜ ≥1 Degree of possibility of SPI

˜ ≤1 Degree of possibility of SPI

1

db1

 ˜ ≥1 = supx≥1 μ ˜ x T SPI SPI

=0

 ˜ ≤1 = supx≥1 μ ˜ x T SPI SPI

=1

Behind the schedule

2

cb1bd

 ˜ ≥1 = supx≥1 μ ˜ x T SPI SPI

=

 ˜ ≤1 = supx≥1 μ ˜ x T SPI SPI

=1

Approximately behind the schedule

3

bb1bc

 ˜ ≥1 = supx≥1 μ ˜ x T SPI SPI

=1

 ˜ ≤1 = supx≥1 μ ˜ x T SPI SPI

=1

On the schedule

4

ab1bb

 ˜ ≥1 = supx≥1 μ ˜ x T SPI SPI

=1

 ˜ ≤1 = supx≥1 μ ˜ x T SPI SPI

=

5

aN1

 ˜ ≥1 = supx≥1 μ ˜ x T SPI SPI

=1

 ˜ ≤1 = supx≥1 μ ˜ x T SPI SPI

=0

d−1 d−c

the activities of the ‘Hospital Sampling’ work package. In this project, it was required to collect 600 hospital samples. These six activities lay over a period of 12 weeks which the Table 5 shows the planned values (PV) and actual costs (AC) up to the data date, i.e. week 6. The information regarding activity progress and activity budget at completion (BAC) are brought in Table 6. Assume here we would like to answer the following questions. • •

˜ and ES ˜? What are the total EV ˜ ˜ and SPI ˜ t? What are the status of the CPI , the SPI

Graphical Description

Decision Making

1−a b−a

Approximately ahead of the schedule

Ahead of the schedule

As Table 8 shows, the progress of all activities is stated using the linguistic terms. These linguistic terms are transformed into fuzzy numbers (F˜ i ) using Table 1 and have brought in Table 7. ˜ of each activity is According to the BAC of each activity, the EV calculated using Eq. (2). Using Eq. (4) we can find the total EV of the work package, i.e. the total EV for all of the four activities: 4

˜ = ∑ EV ˜i = ½1630 1980 2420 2780 EV i=1

Table 5 The PV and the AC of the example. Week

1

2

3

4

5

6

7

8

9

10

11

12

PV AC

100 200

400 500

750 1000

1200 1500

1700 2000

2300 2800

2950 –

3600 –

4150 –

4650 –

4850 –

5000 –

L.M. Naeni et al. / International Journal of Project Management 29 (2011) 764–772 Table 6 The information of work package of the example.

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Table 8 ˜ t calculations. The summary of EAC

Activity

BAC ($)

Progress

i

Ei

Ni

Ei − PVNi

PVNi + 1 − PVNi

ESi = Ni +

1. Filling questionnaire 2. Checking-up people 3. Getting sample 4. Testing sample 5. Analyzing sample 6. Refining sample

800 1000 500 1200 900 600

Very high More than half More than half Less than half Very low Not started

1 2 3 4

1630 1980 2420 2780

4 5 6 6

430 280 120 480

500 600 650 650

4.86 5.47 6.18 6.74

Ei −PVNi PVNi + 1 −PVNi

As Table 7 illustrates, the total PV and the total AC for the work package at the data date, i.e. week 6 are 2300$ and 2800$, respectively. SPI and the CPI of this work package are interpreted as follows.

˜ = ½4 86 5 47 6 18 6 74 . Following these calculations, ES ˜ Table 8 also shows how EAC t should be calculated in the developed model. ˜ t (Eq. (12)) and EAC ˜ t (Eq. (15) given To derive SPI ˜ PF = SPI t ) we have

˜ ˜ = EV = 1630 1980 2420 2780 = ½0 58 0 71 0 86 0 99 CPI AC 2800 2800 2800 2800

˜t= SPI



˜ According  to Table 3 (see scenario 1), T CPI ≥1 = 0 and ˜ T CPI ≤1 = 1. Thus the work package is behind the budget. ˜ ˜ = EV = 1630 1980 2420 2780 = ½0 71 0 86 1 05 1 21 SPI PV 2300 2300 2300 2300

 ˜ Scenario  3 in Table 4 implies that T SPI ≥1 = 1 and ˜ ≥1 = 1, so the work package is on the schedule. These T SPI results imply that this work package is behind the budget and on the schedule. Now, we would like to estimate the completion cost of the work package. Given that the trend affecting the CPI is kept fixed during the rest of the project, the completion cost of the work package is estimated as follows (According to Table 8, the total BAC for all activities is 5000$). ˜ = BAC = BAC × AC BAC × AC BAC × AC BAC × AC EAC ˜ E4 E3 E2 E1 CPI = ½5036 5785 7071 8589

˜, each of the four members of the To calculate the ES ˜ is projected on the baseline and forms a member trapezoidal EV of the trapezoidal fuzzy number ES (Eq. 11). For instance, the calculations behind ES1 are: N1 = 4 as PV4 b E1 = 1630 b PV5 thus   E1 −PV4 1630−1200 ES1 = N1 + =4+ 1700−1200 PV5 −PV4

= 4 86

Table 7 The percent complete and the EV of activities. Activity(i)

BAC ($)

Progress

F˜ i

˜i EV

1 2 3 4 5 6

800 1000 500 1200 900 600

Very high More than half More than half Less than half Very low Not started

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

[640,720,800,800] [500,600,700,800] [250,300,350,400] [240,360,480,600] [0,0,90,180] [0,0,0,0]

=

˜ ES

AD

=

½

4 86

=6

5 47

=6

6 18

=6

6 74

=6

= ½0 81 0 91 1 03 1 12 ˜ t = 6 + 12−6 74 6 + 12−6 18 6 + 12−5 47 6 + 12−4 86 EAC 1 12 1 03 0 91 0 81 = ½10 68 11 64 13 17 14 81

6. Conclusion Measuring the earned value (EV) is one of the first stages in implementing the EV management. There are some different techniques to measure the EV. The Percent Complete is the simplest and the most implemented technique for measuring the EV; however it has the disadvantage of using subjective judgments to describe the percent of the completed work. On the other hand, the fuzzy estimates of both the completion cost and the completion time can assist project managers to estimate the future status of the project in a more robust and a more reliable way. Incorporating the fuzzy principles into the EV calculations does not have the limitations of the previous deterministic EV models and it can model the project statue closer to the reality. In this paper, we used the linguistic terms and fuzzy approach for measuring EV. We explained thoroughly how the fuzzy EV indices and estimates can be developed and can be interpreted. In the interpretation of the fuzzy indices we benefited from evaluating the degree of possibility of a fuzzy number in taking different values. Real-life applications of the proposed fuzzy EV have motivated us to continue the research on the application of the fuzzy theory in the EV calculations and interpretations. As future research directions, we are working on the generalization of the proposed method, and on other fuzzy number ranking methods to interpret the indices and estimates. We would also be interested in developing more comprehensive EV models by using the fuzzy theory. Acknowledgement The authors would like to thank Mr. Hamed Keyvanfar for his comments on the final draft of this paper.

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L.M. Naeni et al. / International Journal of Project Management 29 (2011) 764–772

References Anbari, F., 2003. Earned value project management method and extensions. Project Management Journal 34 (4), 12–23. Bortolan, G., Degani, R., 1985. A review of some methods for ranking fuzzy subsets. Fuzzy Sets and Systems 15 (1), 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 (2), 136–144. Dubois, D., Prade, H., 1980. Fuzzy Sets and Systems Theory and Applications. Academic Press, Inc. Fleming, Q.W., Koppelman, J.M., 2005. Earned Value Project Management3rd 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, pp. 13–23. Summer. Henderson, K., 2004. Further developments in earned schedule. The measurable news, pp. 15–22. Spring 2004. Jacob, D.S., 2003. Forecasting project schedule completion with earned value metrics. The Measurable News 1, 7–9 2003, Spring. Jacob, D.S., Kane, M., 2004. Forecasting schedule completion using earned value metrics revisited. The Measurable News 1, 11–17 Summer.

Kim, E., Wells, W.G., Duffey, M.R., 2003. A model for effective implementation of Earned Value Management methodology. International Journal of Project Management 21 (5), 375–382. Lipke, W., 1999. Applying management reserve to software project management. Journal of Defense Software Engineering 17–21 March. Lipke, W., 2003. Schedule is different. The Measurable News 31–34 Summer 2003. Lipke, W., 2004. The probability of success. The Journal of Quality Assurance Institute, January 14–21 January 2004. 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 (4), 400–407. Zadeh, Lotfi, 1965. Fuzzy sets. Information and Control 8, 338–353. PMI, 2000. Project Management Body of Knowledge (PMBOK). Project Management Institute. PMI, 2004. Project Management Body of Knowledge (PMBOK). Project Management Institute. PMI, 2005. Practice Standard for Earned Value Management. PMI Publication. Vandevoorde, S., Vanhoucke, M., 2005. A comparison of different project duration forecasting methods using earned value metrics. International Journal of Project Management 24 (4), 289–302.