## 1.INTRODUCTION

Project management is the process of planning and guiding a project towards specific objectives within certain time, cost and quality. Project management consists planning, organizing, supervising and guiding which its main objective is to deliver specific and expected results within an agreed budget before a deadline. In other words, project management is the application of process, method, knowledge, skills and equipment to meet project requirements. Project planning and control are the most powerful tools used in project management.

Earned Value management (EVM) as a project management methodology, was created within the United States Defense Department in the 1960s and is one of the most widely accepted control system in project management which helps project managers to measure project performance.

Also, EVM has been known to follow-up both time and cost, the majority of the research has been focused on the cost data (Fleming and Koppelman, 2005; Vandevoorde and Vanhoucke, 2006).

Experts have always criticized the behavior of Schedule Variance (SV) and Schedule Performance Index (SPI) and the way they are interpreted from variety of aspects. According to Vandervoorde and Vanhoucke (2006) SV is measured by cost units not by time-based ones. According to the following issue it would be difficult to understand its behavior and also it may cause misinterpretation. The second criticize to this index is that when SV = 0 or SPI = 1, interpretation of the project status is complicated, because of two different types of interpretation; one will be that the activity is done or it is preceded according to the plan; The other critic is related to the behavior of SPI and SV at the end of the project, in which SV returns to zero and SPI converges to one which declares proper function of the project, even if the project is delayed.

In conclusion, SPI and SV won’t be valid and reliable since a definite time toward the end of the project. So these indices lose their estimation ability usually in the one third final part of the project. This behavior will be regarded as a major problem for the project managers since they won’t be able to rely on these metrics in the most critical period of the project.

The following issue makes severe problems in delayed projects, hence it is not possible to claim that project is behind the schedule because SV will return to zero and SPI returns to 1 (Henderson, 2003).

So, Fleming and Koppelman (2004) suggest that schedule Performance Index should be applied just as a warning mechanism and not as a real tool to analyze how the project is performed with regard to schedule.

Walt Lipke (2003) in order to eliminate the shortcomings of EVM introduced the concept of Earned Schedule (ES) technique which leads to computing change of EVM schedule indicators. Also, ES and presented index known as SPI (t) which is a better schedule performance measure compared to SPI, but using cost-based data decreased the reliability (Vanhoucke *et al*., 2015).

In this regard, Khamooshi and Golafshani (2014) published a new approach known as DM for project schedule performance management which eliminates the use of cost data in the schedule’s context. Its foundation lies in the exclusive usage of time-based data for the generation of physical Progress indicators (Vanhoucke *et al*., 2015).

However, EDM, EVM and ES techniques activities are considered deterministic, nature of some activities are indefinite, mostly because the data regarding the activities come from people’s judgments which carry some degree of uncertainty. For this reason, interpreting and calculating this uncertainty would cause better performance measurement and extend EDM applicability in real-life and indefinite conditions.

In this regard, there are studies which paid attention to uncertainty in project management. Naeni *et al*. (2011) developed a new fuzzy-based EVM technique to measure project performance and progress under uncertainty. Dehabadi *et al*. (2014) in order to deal with the ambiguity and fuzziness of real data in project, proposed a theoretical framework to estimate future performance of project regarding the past relative information which benefits from fuzzy regression (FR) models. Ponz-Tienda *et al*. (2012), considering duration, cost and production and alternatives in the scheduling between the earliest and latest times, present a proposal for project scheduling and control by applying fuzzy EV and their results, suggest that: “different possible schedules and the fuzzy arithmetic provide more objective results in uncertain environments than the traditional methodology”.

So, in this paper, a fuzzy approach is proposed to measure project performance using EDM technique under uncertainty condition. EDM technique has been introduced in section 2. It is followed by description of fuzzy theory and applying fuzzy approach to Earned Duration Management in section 3, section 4 is dedicated to develop fuzzy performance and estimation indices and it is followed by interpretation of proposed fuzzy indices in section 5 and eventually to clarify the proposed indices, a real case study is employed in section 6.

## 2.THE EARNED DURATION MANAGEMENT

The Earned Duration Management is a novel approach which use time-based data exclusively and decouples duration and cost for the purpose of performance management and measurement, so it is the counterpart or complement of EVM and takes care of duration and schedule management of any project (Khamooshi and Golafshani, 2014), therefore schedule performance metrics become free from any dependency on planned cost values and are no longer influenced by them (Vanhoucke *et al*., 2015; Soleymani *et al*., 2015).

This technique introduces time-based indices such as Total Planned Duration (TPD), Total Earned Duration (TED) and Total Actual Duration (TAD); it can be said that they are equivalent twin of PV, EV and AC for EVM.

Batselier and Vanhoucke (2015) concluded that EDM(t) certainly proves to be a valid methodology for forecasting project duration, as it can compete with — and potentially improve — the currently most recommended methodology of ESM.

To keep the paper short, a list of indices, formulas and their description proposed in the EDM technique, is presented in Appendix. A, which more details can be found in “EDM: Earned duration Management, a new approach to schedule performance management and measurement” (Khamooshi and Golafshani, 2014).

## 3.APPLYING FUZZY APPROACH TO EARNED DURATION MANAGEMENT

Fuzzy sets theory first introduced by Lotfi Zadeh (1965) applied in order to manage ambiguity and impreciseness in reality. So, using linguistic term, the fuzzy theory can model and treat the uncertainty in such areas. With the aim of reaching this goal, fuzzy theory employs different types of numbers with certain membership function (Salari *et al*., 2014).

In general, the related membership functions of a trapezoidal fuzzy number, for instance:

$\tilde{A}=\left[{a}_{1},\hspace{0.17em}{a}_{2},\hspace{0.17em}{a}_{3},\hspace{0.17em}{a}_{4}\right]$ is defined as below:(1)

The trapezoidal fuzzy number becomes triangular fuzzy number if *a*_{2} = *a*_{3}. in this paper due to easiness in calculation, a triangular fuzzy number is also represented as a trapezoidal fuzzy number [*a*_{1}, *a*_{2}, *a*_{3}, *a*_{4}] or [*a*_{1}, *a*_{3}, *a*_{3}, *a*_{4}].

The basic operations of these two fuzzy numbers are presented as follows (Zadeh, 1965):

Assume *r* ≥ 0 is a real number and $\tilde{A}$ and $\tilde{B}$ are two trapezoidal fuzzy numbers with four numbers:

$\begin{array}{l}\tilde{A}+\tilde{B}=\left({a}_{1}+{b}_{1},\hspace{0.17em}{a}_{2}+{b}_{2},\hspace{0.17em}{a}_{3}+{b}_{3},\hspace{0.17em}{a}_{4}+{b}_{4}\right)\\ \tilde{A}-\tilde{B}=\left({a}_{1}-{b}_{1},\hspace{0.17em}{a}_{2}-{b}_{2},\hspace{0.17em}{a}_{3}-{b}_{3},\hspace{0.17em}{a}_{4}-{b}_{4}\right)\\ \tilde{A}\times \tilde{B}=\left({a}_{1}\times {b}_{1},\hspace{0.17em}{a}_{2}\times {b}_{2},\hspace{0.17em}{a}_{3}\times {b}_{3},\hspace{0.17em}{a}_{4}\times {b}_{4}\right)\\ \tilde{A}\xf7\tilde{B}=\left(\raisebox{1ex}{${a}_{1}$}\!\left/ \!\raisebox{-1ex}{${b}_{4}$}\right.,\hspace{0.17em}\raisebox{1ex}{${a}_{2}$}\!\left/ \!\raisebox{-1ex}{${b}_{3}$}\right.,\hspace{0.17em}\raisebox{1ex}{${a}_{3}$}\!\left/ \!\raisebox{-1ex}{${b}_{2}$}\right.,\hspace{0.17em}\raisebox{1ex}{${a}_{4}$}\!\left/ \!\raisebox{-1ex}{${b}_{1}$}\right.\right)\\ \tilde{A}\times r=\left({a}_{1}\times r,\hspace{0.17em}{a}_{2}\times r,\hspace{0.17em}{a}_{3}\times r,\hspace{0.17em}{a}_{4}\times r\right)\end{array}$

The proposed method, apply where the amount of the work required to perform the activities are unknown or uncertain. For instance, in Road construction projects, in order to achieve the desirable level, if excavation is required, it should be continued till we reach the ground with the sufficient persistence; also, if there is a need of embankment, first the soil must be removed till the suitable depth is gained and then start the operation.

In many medical research cases, because of the clinical experiments, the exact amount of work required to derive scientific conclusions is unknown (Naeni *et al*., 2011).

Assume that an activity progress can’t be stated deterministic. In this regard, Linguistic terms can help, so it can state as “less than half”, “very high”, etc. Linguistic terms, make it easier to figure out the activity progress by answering this question: “How much of the activity is completed?” it is obvious that these linguistic terms should be transformed to a number first and then applied on the EDM technique. For this purpose, according to Naeni *et al*. (2011), we assign a membership function to this linguistic term.

Figure1 and Table 1 show this process:

For example, the linguistic term “more than half” equals to the fuzzy number [0.5, 0.6, 0.7, 0.8].

So, if an activity progress can’t be stated deterministic, the linguistic term can help to state it. Afterward, it should be transformed to a mutual number using Table 1 to apply on EDM technique.

Now we can modify the EDM metrics to demonstrate the new values (fuzzy numbers).

## 4.DEVELOPMENT OF FUZZY PERFORMANCE AND TIME ESTIMATION INDICES

Assume that ${\tilde{P}}_{i}$ is the fuzzy activity progress of the activity i, i.e.(2)

According to Appendix A, earned duration of a scheduled activity is calculated using Eq. (3)

Note that *BPDi* is the Baseline Planned Duration assigned to activity *i*.

To derive Total Earned Duration (TED) in each period, we should sum up all ${\tilde{ED}}_{i}$ for $i=1,\hspace{0.17em}\cdots ,\hspace{0.17em}n$, n (n is the total number of project activities):

In this equation, the *ED _{i}* can be both fuzzy numbers and deterministic numbers. Now we can express all related indices as fuzzy numbers.

Khamooshi and Golafshani (2014) defined performance measurements on two levels named as: Activity level and Project Level.

To keep the paper short, some of activity level performance and estimation indices are proposed here; it is obvious that development of others is the same too.

Earned Duration (ED (t)) at any point in time in deterministic form, is the duration corresponding to Total Earned Duration (TED) on Total Planned Duration Scurve. Khamooshi and Golafshani (2014) portray a conceptual graph of ED (t) (see Figure 2).

Eq. (5) show $\tilde{ED\left(t\right)}$ :

Note that the calendar unit represents the unit in which time instant t is measured.

Duration Performance Index (DPI) compares the project’s progress made with the time passed and shows the achievement of project target completion date, i.e. $DPI=\frac{ED\left(t\right)}{AD}$.

$\tilde{DPI}$ Is calculated in Eq. (6)

Also, Earned Duration Index (EDI) compares the overall actual achievement with planned achievement, i.e. $EDI=\frac{TED}{TPD}$

$\tilde{EDI}$ Is calculated in Eq. (7)

In addition, EDM technique has indices which can estimate the completion duration of the project.

According to Appendix. A, a common formula to estimate duration of the project assuming the future trend of the project remain intact, is to divide BPD by the Dura tion

Performance Index (DPI), i.e. $EDAC=\frac{BPD}{DPI}$.

As in the previous section proposed DPI is a fuzzy number, EDAC become a fuzzy number too:

In Eq. (8), assume that $\tilde{DPI}$ would be constant for the remainder of the project.

## 5.INTERPRETATION OF PROPOSED EDM FUZZY INDICES

The developed fuzzy indices should be interpreted for decision making. It means that we should transform these fuzzy numbers in order to compare them with planned values.

In this regard, we can use various methods of comparing fuzzy numbers. These methods are commonly divided into two categories. First category is defuzzification method and the other is comparing fuzzy numbers by preference relation. Wang (2015) believed that defuzzification, because of losing fuzzy messages, is simpler than ranking fuzzy numbers (fuzzy pair-wise comparison). Whereas, fuzzy pair-wise comparison is complicated, however it reserves fuzzy messages. Also, Yuan (1991) supposed that a fuzzy ranking method had to present preference relation in fuzzy terms. So out of different methods in the literature (Wang, 2015; Lee and Li, 1988; Choobineh and Li, 1993; Adamo, 1980; Requena *et al*., 1994; Fortemps and Roubens, 1996), ranking fuzzy numbers based on the relative preference relation introduced by Wang (2015) is implemented in this paper, however it is complicated but it reserves fuzzy messages.

According to Wang (2015), assume A and B two trapezoidal fuzzy numbers, where:

$\tilde{\text{A}}=\left({\text{a}}_{\text{1}}\text{,}\hspace{0.17em}{\text{a}}_{\text{2}}\text{,}\hspace{0.17em}{\text{a}}_{\text{3}}\text{,}\hspace{0.17em}{\text{a}}_{\text{4}}\right)\hspace{0.17em}and\hspace{0.17em}\tilde{\text{B}}=\left({\text{b}}_{\text{1}}\text{,}\hspace{0.17em}{\text{b}}_{\text{2}}\text{,}\hspace{0.17em}{\text{b}}_{\text{3}}\text{,}\hspace{0.17em}{\text{b}}_{\text{4}}\right)$, then:(9)

$\Vert T\Vert $ Is the extending difference between $\tilde{A}$ and $\tilde{B}$ and calculated as follow:(10)

$\begin{array}{l}\begin{array}{cc}{t}_{1}^{+}=\mathrm{max}\left\{{a}_{1},{b}_{1}\right\},& {t}_{2}^{+}=\mathrm{max}\left\{{a}_{2},{b}_{2}\right\}\end{array},\\ \begin{array}{cc}{t}_{3}^{+}=\mathrm{max}\left\{{a}_{3},{b}_{3}\right\},& {t}_{4}^{+}=\mathrm{max}\left\{{a}_{4},{b}_{4}\right\}\end{array},\\ \begin{array}{cc}{t}_{1}^{-}=\mathrm{max}\left\{{a}_{1},{b}_{1}\right\},& {t}_{2}^{-}=\mathrm{min}\left\{{a}_{2},{b}_{2}\right\}\end{array},\\ \begin{array}{cc}{t}_{3}^{-}=\mathrm{max}\left\{{a}_{3},{b}_{3}\right\},& {t}_{4}^{-}=\mathrm{min}\left\{{a}_{4},{b}_{4}\right\}\end{array},\end{array}$

Whereas the comparison is made against a deterministic number, to apply Wang (2015) method to Eq. (6)- (8):

Assume *X* ≥ 0 is a real number, $\tilde{A}$ is preferred to $X\hspace{0.17em}If\hspace{0.17em}{\mu}_{p}\left(\tilde{A},\hspace{0.17em}X\right)>\frac{1}{2}$.

Note that $\tilde{DPI}$ and $\tilde{EDI}$ should compare against value 1, since $\tilde{EDAC}$ compare against the planned Value.

Respectively, Table 2 and Table 3 interpret the comparison of $\tilde{EDI}$ and $\tilde{DPI}$ against value 1.Table 4

The Baseline Planned Duration (BPD) is the total authorized duration assigned to the project. Project expected to complete within this total duration. Beside this, we define Schedule Reserve (SR) which is the duration used in unexpected situation. Thus, the Final Authorized Date (FAD) of project completion calculated like this:

$\text{FAD=BPD+SR}$

In this regard, EDAC interpretation is:

## 6.CASE STUDY

In this section, a case study from a road construction industry is illustrated to show that how the development approaches can be used for a real project. The project consists of 8 work packages. The baseline planned duration for this project is 22 weeks and for simplicity we mean both activities and work packages by “activity.”

Table 5 Demonstrates the Total Planned Duration (TPD) and Total Actual Duration (TAD) up to week 6 which is the data date. Work package codes, names, duration and progress information of the project are brought in Table 6.

First of all, activity progress which is stated using linguistic terms should be transformed to fuzzy numbers $\left({\tilde{P}}_{i}\right)$, then using Eq. (3) the Earned duration of each activity is calculated.

For instance, ${\tilde{ED}}_{2}$ is:

${\tilde{ED}}_{i}$ And ${\tilde{P}}_{i}$ of each activity is presented in Table 6.

According to Eq. (4) and Table 7, Total Earned Duration (TED) for all activities up to week 6 equals to:

In order to calculate fuzzy performance indices, $\tilde{ED\left(t\right)}$ is needed:

According to Eq. (5):

Applying Eq. (5) for $\tilde{ED\left({t}_{2}\right)},\hspace{0.17em}\tilde{ED\left({t}_{3}\right)}\hspace{0.17em}\text{\&}\hspace{0.17em}\tilde{ED\left({t}_{4}\right)}$ will result in:

So according to Eq. (6) and Eq. (7), $\tilde{DPI}$ and $\tilde{EDI}$ of this project up to data date are calculated as follow:

To interpret the fuzzy numbers of above indices, Eq. (11) and Eq. (12) are used:

${\mu}_{p}\left(\tilde{EDI},1\right)<\frac{1}{2}$ thus, Project achieves less amount of work in comparison to Plan according to Table 2.

Also, same as above ${\mu}_{p}\left(\tilde{DPI},1\right)$ is calculated:

$\Vert \tilde{DPI},1\Vert =\Vert {T}_{2}\Vert =0.17\hspace{0.17em}\text{and}\hspace{0.17em}\text{}{\mu}_{p}\left(\tilde{DPI},1\right)=0.27<\frac{1}{2}$ so there is Worse Performance Compare to Plan in this project (according to Table 3).

In addition, to calculate fuzzy estimated duration at completion ($\tilde{EDAC}$) of this case (assuming $\tilde{DPI}$ kept fix duration the rest of the project) according to Eq. (8):

The BPD of the project is 22 weeks, assuming SR for this case is 5 weeks; thus, the Final Authorized Date (FAD) of project completion equals to:

According to Eq. (11) and Eq. (12):

According to Table 4: *BPD* < $\tilde{EDAC}$ ≤ *FAD*, so Project needs SR to complete within FAD and the project managers should consider corrective action to reduce and adjust project duration.

## 7.CONCLUSION

There are various methods such as EVM, ES and others, which are used to assess measure and quantify project performance and progress. In this regard, one of the most recent techniques which named EDM wipe out the use of cost data, so schedule performance measures are calculated according to time-based indices. However, all these methods are only applicable in deterministic situations but in reality, for instance expressing progress of activities comes from people’s judgments which include degree of uncertainty. Encountering this problem, this paper applies a novel fuzzy approach to measure project performance and estimation using EDM technique. This approach derives development of new fuzzy indices which are capable to measure project performance under uncertainty. In order to interpret these new fuzzy metrics, a fuzzy ranking method based on the relative preference relation is employed which is eventually clarified by the utilization of proposed fuzzy indices in a real road construction project. Obtained results elaborate that not only, this sample project is behind the schedule, but also, in comparison to the plan, they have gained worse performance. Therefore, the project won’t be able to be completed within the authorized duration and required corrective actions. It is obvious that the proposed approach provides more realistic results and developing this approach may derive novel solutions for uncertainty in project management. We would also be interested in implementation of various fuzzy ranking methods for the proposed approach.