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A comparative analysis of learning curves: Implications for new technology

implementation management

Malgorzata Plaza

a

, Ojelanki K. Ngwenyama

a,b,*

, Katrin Rohlf

c

a

Institute for Innovation and Technology Management, Ryerson University, 55 Dundas Street West, 9 Floor, Room 3-089, Toronto, Ontario, Canada M5B 2K3

b

Aarhus School of Business, Aarhus University, Denmark

c

Department of Mathematics Ryerson University, Toronto, Canada

article info

Article history:

Received 8 September 2007

Accepted 5 January 2009

Available online xxxx

Keywords:

Project management

Information systems implementation

Learning curve models

Organization learning

abstract

New technology implementation projects are notoriously over time and budget resulting in signiﬁcant

ﬁnancial and strategic organizational consequences. Some argue that inadequate planning and manage-

ment, misspeciﬁcation of requirements, team capabilities and learning contribute to cost and schedule

over runs. In this paper we examine how learning curve theory could inform better management of

new technology implementation projects. Our research makes four important contributions: (1) It pre-

sents a comparative analysis of learning curves and proposes how they can be used to help ERP imple-

mentation planning and management. (2) Based on empirical data from four ERP implementation

projects, it provides illustrations of how managers can apply the curves in different project situations.

(3) It provides a theoretical basis for empirical studies of learning and ERP (and other IT) implementations

in different organizational settings. (4) It provides empirical justiﬁcation for the development of learning

curve theory in IT implementation.

Crown Copyright Ó2009 Published by Elsevier B.V. All rights reserved.

1. Introduction

Cost and schedule overruns during new technology implemen-

tation projects are well documented in academic and practitioner

journals. Recent studies (Robbins-Gioia, 2002; The Standish Group,

2004) ﬁnd that more than 70% of ERP implementations are over

schedule and budget. Much of the literature aimed at ‘ﬁxing’ the

ERP implementation problem focuses on prescribing ‘best practice’

for successful ERP implementation (Davenport, 1998; Markus

et al., 2000; Besson and Rowe, 2001). Some of the prescriptions

encourage project managers to: (a) adopt standard business pro-

cesses to ﬁt with the ERP software (Markus et al., 2000; Palanisw-

amy and Frank, 2000; Sumner, 2000); (b) avoid customizing the

software (Parr and Shanks, 2000; Mabert et al., 2001; Murray

and Cofﬁn, 2001); and (c) provide appropriate user training (Bingi

et al., 1999; Holland and Light, 1999; Al-Mudimigh et al., 2001).

Other researchers offer frameworks to assist managers in deﬁning

and analyzing critical success factors (Akkermans et al., 1999; Bingi

et al., 1999; Holland and Light, 1999; Nah et al., 2001), and project

risk factors (Sumner, 2000). Still others suggest the building of in-

tra-organizational coalitions to support ERP implementation pro-

jects (Pozzebon and Pinsonneault, 2005). However, there is now

growing acknowledgment that team training/learning and im-

proved project management methods are important to ERP imple-

mentation success (Bingi et al., 1999; Holland and Light, 1999;

Kumar and van Hillegersberg, 2000; Willcocks, 2000; Hong and

Kim, 2002; Verville and Halingten, 2002).

Recent research has identiﬁed a relationship between project

team capabilities and IT implementation effectiveness. For exam-

ple, Chatzoglou and Macaulay (1996) found that project team capa-

bilities are the single most important factor affecting the timelines

of software implementation projects. A recent study by Karlsen

and Gottschalk (2003) also found that knowledge transfer is signif-

icantly related to IT project success. Dixon (2000) found that

knowledge transfer among teams enables problem solving on IT

projects. Several researchers have also pointed out that inaccurate

predictions of team learning performance often lead to conse-

quences of project creep and cost escalation (Keil et al., 1995; Chat-

zoglou and Macaulay, 1996; Callaway, 1999; Barry et al., 2002;

Depledge, 2003; Wiegers, 2003; Wallace and Keil, 2004). ERP

implementations projects are especially vulnerable schedule over-

runs because: (1) ERP software is complex and, in most cases, the

managers of organizations implementing these systems often have

little or no prior experience with them; (2) the scale of the imple-

mentation is often larger in scope than any previous IT implemen-

tation in the organization; (3) the organization’s own IT specialists

0377-2217/$ - see front matter Crown Copyright Ó2009 Published by Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2009.01.010

*Corresponding author. Address: Institute for Innovation and Technology

Management, Ryerson University, 55 Dundas Street West, 9 Floor, Room 3-089,

Toronto, Ontario, Canada M5B 2K3. Tel.: +1 416 979 5000/4203; fax: +1 416 979

5249.

E-mail addresses: mplaza@ryerson.ca (M. Plaza), ojelanki@ryerson.ca (O.K.

Ngwenyama), krohlf@ryerson.ca (K. Rohlf).

European Journal of Operational Research xxx (2009) xxx–xxx

Contents lists available at ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier.com/locate/ejor

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often have little knowledge of the ERP system and project must be

augmented by specialized external consultants; (4) the schedule

for completion of the ERP implementation is often tight due to

competitive pressures on the organization. Seen from this perspec-

tive, the management project team learning capabilities is an

important dimension of effective project management. Unfortu-

nately, although it is accepted that the team’s learning rate and

capabilities inﬂuence its performance and productivity, little re-

search has been conducted on this issue (Barry et al., 2002).

In this paper we investigate the potential of learning curves as

a method for planning and management of ERP implementation

projects. This paper extends the work of Ngwenyama et al.

(2007) who proposed a learning curve model for managing orga-

nizational software upgrades cycles. We focus on three key ques-

tions: (1) How can managers achieve effective project team

learning that is so critical to the success of ERP implementation?

(2) How can managers develop viable project schedules that take

into account the learning curve of the project team? and (3) How

can managers identify and dynamically adjust project plans and

resources in response to contingencies during ERP implementa-

tion. Our discussion of these issues unfolds as: in Section 2we

discuss the linkage between learning and project performance,

and the challenges of managing ERP projects. In Section 3we dis-

cuss the basic and assumptions of two learning curve models, the

S-curve and the simpliﬁed exponential curve. In Section 4we

present a comparative analysis of the implications of using both

curves using empirical data from four ERP implementation cases

from three companies. Finally in Section 5we conclude with

implications for new technology implementation theory and

practice.

2. The learning and performance relationship

The relationship between learning and project performance was

ﬁrst formally deﬁned by Wright (1936). Since then, several

researchers have investigated and modeled this relationship. To-

will (1985) studied on-the-job (or industrial) learning and devel-

oped a set of parameters for measuring learning and productivity

gains in industrial projects. Terwiesch and Bohn (2001) investi-

gated the cost and beneﬁts of learning during ramp-up production

of new products. Their primary interests were the cost of speeding-

up learning and experimentation, and the likelihood and cost of

product defects as teams progressed to full production.

Although on the surface it might appear that new product man-

ufacturing and ERP implementation projects are different, they

share important features. In both cases the project teams must

learn new production methods and work with designs of end prod-

ucts of which they may have little knowledge. In both cases the

teams need to learn from experience in spite of whatever initial

training they may receive. Further, the cost of product defects

could be quite signiﬁcant and threatening to the economic viability

of the company. From this perspective, ERP implementation pro-

jects can be viewed as limited-time endeavors that can be planned

and managed similarly to some new product manufacturing pro-

jects. One key difference is that the ERP project team is cross-dis-

ciplinary, comprising technical and business experts from various

organizational functions, as well as external consultants hired to

support the endeavor (Davenport, 1998;Boyer, 2001; Baccarini

et al., 2004). Team members of different backgrounds, business

disciplines and interests must learn and work together if the

implementation project is to succeed (Fedorowicz et al., 1992;

Edmondson et al., 2003). Careful selection of the team members,

a good project plan, high quality training in the ERP software and

implementation practices along with strong top management sup-

port are also essential to success (Davenport, 1998; Boyer, 2001;

Baccarini et al., 2004).

An important factor in successful ERP implementation projects

is early team learning. New implementation teams require signiﬁ-

cant initial classroom training on the ERP software followed by

experiential learning on conﬁguring the software in a sandbox set-

ting (Robey et al., 2002). And since team members play different

roles on the project, they need to develop cross-functional skills

(Boyer, 2001; Rocheleau, 2006). Further, many issues challenge

team learning and performance during implementation. Software

features may not be well understood; some members leave the

team and new ones ﬁll their places (Chambers, 2004; Pendharkar

and Subramanian, 2006). A wide range of hygiene factors, such

as boredom, fatigue, distractions, can also impede individual and

team learning and performance (Eason, 1988; Adler and Clark,

1991). Thus a key problem for project managers is to understand

the rate of project team learning in order to effectively plan and de-

velop the project schedule. Predicting the impact of team learning

in complex projects is essential to effective project management

but it can be quite challenging (Eden et al., 1998; Ellis and Shpiel-

berg, 2003).

3. Modeling learning and performance

Since Wright, various learning curve models have emerged, but

the two main curves used for cost estimation and productivity

assessment are the S-curve and the exponential progress curve

1

(Yelle, 1979; Towill, 1985; Argote and Epple, 1990; Pananiswami

and Bishop, 1991; Teplitz, 1991; Badiru, 1992; Mosheiov and Sidney,

2003). The S-curve conceptualizes performance improvement as

function of practice, with the most dramatic improvements taking

place at the beginning of the learning process. According to Arrow

(1962), knowledge increases with experience and the process of

learning is a product of experience. In an S-curve, either time or a

cumulative output can be chosen as an independent variable and

inﬂuences of experience carry-over, cessation of learning and a start

up effect are considered (Howell, 1990). Interestingly, an exponen-

tial curve is the one most commonly used to track performance in

technology related projects (Butler, 1988; Teplitz, 1991; Teplitz

and Amor, 1993; Jovanovic et al., 1995; Jackson, 1998; Chambers,

2004; Dardan et al., 2006; Pendharkar and Subramanian, 2006).

The logistic curve is similar to the S-curve but ignores the start up

effect. The rationale for using an exponential model is that it ade-

quately describes the performance improvement during the experi-

ential learning phase. The logistic model is robust and has a wide

range of applicability. Since in the logistic model output is related

to asymptotic performance, the curve coefﬁcient, k, relates changes

in performance over time to the performance threshold (when learn-

ing is complete) (Yelle, 1979; Edgington and Chen, 2002). In the next

section we examine the relationship between the logistic model and

the S-curve model, and discuss the range of applications of the logis-

tic model and illustrate how it can be used as an analytical tool to

assist project managers in analyzing learning and productivity issues

and their implications for IS project scheduling and duration.

3.1. Basic learning curve concepts

As stated earlier, learning curve models used on technology and

IS projects are often called progress curves or progress functions

(Malerba, 1992). Progress curves model practice and performance,

where practice is represented by units of time (or the number of

times a predeﬁned output is delivered), and performance is mea-

sured as a rate, in which a predeﬁned output is produced (Fedo-

rowicz et al., 1992). In classical form, progress curves depict a

1

Space limitation prevents a detailed discussion of learning curve models; the

interested reader can refer to Badiru (1992) and Yelle (1979).

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rapid increase of early performance then a decline and diminish-

ment to 0. James (1984) reports that variations in performance rel-

ative to practice and experience track the S-curve more closely

when hygiene factors that affect performance and learning such

as boredom, fatigue and distractions are considered.

For our purposes of modeling and analyzing learning and per-

formance in ERP implementation projects we deﬁne an initial team

training exercise and a two phase implementation cycle, compris-

ing conﬁguration and testing/conversion phases (Markus et al.,

2000). Assuming that ERP implementation follows a progress

curve, we can model the relationship between the progress curve

coefﬁcient kand performance. In this case we deﬁne performance

as the rate of completion of a task; for example: (1) the number of

sessions conﬁgured by a single team member in a unit of time, or

(2) the number of transactions completed by a single team mem-

ber in a unit of time. We now deﬁne the basic terms and concepts

we use in modeling the logistic and Scurves:

Deﬁnition of model terms

P

l

(t), P

s

(t) progress functions as functions of time, t

p

t

performance threshold (constant performance function)

represents the optimal performance of the fully trained

and integrated team operating at their peak performance

levels

T

0

planned project duration is the expected time for comple-

tion of the implementation. It is estimated in weeks,

months or fractions of the above depending on which units

is project duration measured in. It is calculated based on

the assumption that performance remains predetermined

and constant during the progression of the project

T

l

,T

s

project durations, calculated in the same units as used for

the planned project duration. They are estimated based on

the assumption that performance on the project is repre-

sented by either P

l

(t)orP

s

(t)

D

T

l

,

D

T

s

extensions of the planned project duration. They are calcu-

lated as the difference between Tand T

0

kprogress curve coefﬁcient

T

0k

knowledge absorption capacity coefﬁcient

mTeam’s initial performance level measured at the begin-

ning of the project and scaled by p

t

D

0

(t),

D

(t) the difference between the logistics and S-curves mea-

sured as the difference in areas underneath those curves

at time tscaled by p

t

3.2. Basic assumptions

We now outline three basic assumptions upon which our anal-

ysis is based; there are planned project duration,team performance

levels and forgetting. Planned or expected project duration is a crit-

ical project parameter that is usually underestimated by project

managers due to uncertainty about (a) the team’s performance

capability and (b) the amount of work required to conﬁgure, test

and install the ERP system. Project duration is often optimistically

estimated with an input from the software vendor, who is often

biased by their interest in selling the software. Assumption 1 be-

low establishes a key link between planned project duration and

the progress curve models.

Assumption 1. If the performance of each team member is at p

t

at

all times during the IS project, then the total time required for the

implementation will be equal to planned duration T

0

.

This assumption implies a fully trained and integrated project

team capable of peak performance. Such a team is seldom available

except from the software vendor or specialized consultants who

have gained competence from repeated implementation of the

software in other ﬁrms. In practice, however, most ERP implemen-

tation projects will require time extensions (deﬁned as

D

T) beyond

the planned duration. Therefore we will relax Assumption 1 to P

s

(t)

when project team performance follows the S-curve and P

l

(t) when

it follows the simpliﬁed logistic model. Team performance will al-

ways fall below the performance threshold p

t

, where tis time

elapsed on the project.

Fig. 1 illustrates the performance outcomes when Assumption 1

is strong and relaxed. When Assumption 1 is valid (strong) the to-

tal amount of work required to complete the project is represented

by the area deﬁned as R

T

0

p

t

dt. When Assumption 1 is relaxed

implementation time is extended by

D

T

l

when logistic curve is

used, and

D

T

s

when S-curve is utilized. However, the total work re-

quired remains unchanged as illustrated by the two integrals

R

T

l

0

P

l

ðtÞdt ¼AreaðLogistics curveÞ;

R

T

s

0

P

s

ðtÞdt ¼AreaðScurveÞ:

(

3.2.1. Factors affecting team performance

Several factors affect team performance including team mem-

bers’ knowledge, practice and experience gained on implementa-

tion projects. These performance changes are measured (the

coefﬁcient k), cumulatively in terms of reduction of time required

to complete various implementation tasks. Other factors inﬂuenc-

ing performance improvements during ERP implementations are:

(1) team cohesion resulting from storming, forming and norming

into the performing stage (Gray and Larson, 2000), (2) internal

team members learning the new ERP software, and (3) external

team members (consultants) learning the business processes of

the organization (Davenport, 1998). Thus for this analysis we make

the following assumption:

Assumption 2. We estimate the progress curve coefﬁcient k

(changes in team performance levels) from the average team

member’s performance and adjust for the team environment.

A key factor contributing to forgetting is the time lag between

training and task execution. This is a situation that does not nor-

mally exist in ERP implementations. Team members are trained

just before the start of the project and the training includes an

experiential component in a sandbox. Further, ERP implementation

projects are carried out under conditions conducive to learning and

performance such as: (1) executive level support and involvement;

(2) expert consulting support and guidance; (3) project schedules

no longer than 18 months; and (4) performance incentives (Daven-

port, 1998; Boyer, 2001; Baccarini et al., 2004; Dong, 2004). Condi-

tions which (Fedorowicz et al., 1992) argue, promote learning and

overcome boredom, fatigue and forgetting. Thus in our analysis we

ignore performance decay due to forgetting:

0

Pl, P s

t

Performance Threshold

Progress Function Pl: Logistic curve

Progress Function Ps: S-curve

Tl

T0

Δ

T

Training

Configuration/ Testing/ Go live

Ts

Δ

T

Fig. 1. Illustration of the model parameters.

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Assumption 3. The impact of forgetting is not considered in either

model.

3.3. Formal description of the progress curve models

We deﬁne a progress function Pfor the average project team

member, with kas the progress curve coefﬁcient, and p

t

the perfor-

mance threshold. The performance growth rate is thus the deriva-

tive

dP

s

dt

expressed by the S-curve equation (3.1)

dP

s

dt ¼kP

s

1P

s

p

t

:ð3:1Þ

The solution to the Eq. (3.1) is

P

s

ðtÞ¼ p

t

1þAe

kt

;ð3:2Þ

where A¼

p

t

p

0

p

0

¼

p

t

p

0

1¼m1 and m¼

p

t

p

0

represents the project

team’s initial performance (i.e. p

0

) measured relative to the perfor-

mance threshold p

t

.

Eq. (3.2) can be further transformed into dimensionless pro-

gress function

P

s

ðtÞ

p

t

¼1

1þAe

kt

:ð3:2

0

Þ

Performance growth rate can also be expressed as the logistics

equation

dP

l

dt ¼kðp

t

P

l

Þð3:3Þ

the solution to which

P

l0

ðtÞ¼p

t

ð1e

kt

Þif P

l

ð0Þ¼0;

P

l

ðtÞ¼p

t

ð1

m1

m

e

kt

Þif P

l

ð0Þ¼p

0

(ð3:4Þ

is again transformed into dimensionless form as

P

l0

ðtÞ

p

t

¼1e

kt

if P

l

ð0Þ¼0;ð3:4:1

0

Þ

P

l

ðtÞ

p

t

¼1m1

me

kt

if P

l

ð0Þ¼p

0

:ð3:4:2

0

Þ

The difference between the logistic and S-curves, is deﬁned by the

difference in areas underneath those curves at time tscaled by p

t

(cf. Fig. 1) will be represented as either

D

0

(t) when the initial con-

dition P

l

(0) = 0 is applied to the logistic curve (3.4.1)

0

,or

D

(t) when

the initial condition P

l

(0) = p

0

is used.

Theorem 1. for proof see Appendix A.When the team reaches an

initial level of performance p

0

¼

p

t

e

before the commencement of the

project (m equals to constant e) it makes no difference which curve is

used to track performance since both the S-curve and logistic curve

become identical in relatively short time.

Theorem 1 explain why both curves have been used success-

fully to track performance on ERP projects when team performance

reaches the level of p

0

equal or close to 30% of the performance

threshold prior to the commencement of the project. We will

now deﬁne Theorem 2 as an extension of Theorem 1 to situations

where the project team did not reach the critical level of perfor-

mance before the commencement of the project. The difference be-

tween the logistics and S-curves measured as the difference in

areas underneath those curves at time tscaled by p

t

(Fig. 2) will

be represented as

D

0

(t) (Eq. (A.2)

0

).

Theorem 2. for proof see Appendix B.When m >e (i.e. p

0

does not

reach 30% of performance threshold), j

D

0

(t)jdepends only on the

coefﬁcient k and parameter m ¼

p

t

p

0

and will reach its maximum as

when implementation time approaches inﬁnity t ?1or as imple-

mentation time equal to critical time t

*

(Eq. (3.5)).

t

¼1

kln p

t

2p

0

p

t

p

0

:ð3:5Þ

The reader should take note that when the team’s initial perfor-

mance is low (i.e. p

0

does not exceed 30% of performance thresh-

old), then the difference between the logistic and S-curves

D

0

(t)

is calculated for the initial condition of P

l

(0) = 0. Fig. 3 provides a

graphic illustration of Theorem 2. On the left plot

D

0

(t) is a func-

tion of very short project duration (depicted in weeks) and on

the right plot a longer project duration (depicted in months). It

0

Pl, Ps

t

Performance Threshold p t

Progress Functions Ps:

S-curve Progress Function Pl, Pl(0)=0

Logistic curve

Fig. 2. The difference between the logistic (initial condition of 0) and S-curves

measured as the difference in areas underneath those curves at time tfor cases

when m=eor m>e.

k=0.8

0

0.5

1

1.5

2

2.5

1

Implementation time t [months]

Δ0(t)

m=3

m=4

m=10

m=14

k=0.8

m=3

m=4

m=10

m=14

0.5

0

Δ0(t)

-0.2 19171513119753

Fig. 3. The difference between the logistic and S-curves as a function of time: measured for short (plot on the left) and long (plot on the right) implementations.

4M. Plaza et al. / European Journal of Operational Research xxx (2009) xxx–xxx

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can be seen from the plots that j

D

0

(t)jreaches its maximum in less

than 6 weeks and becomes smaller as mbecoming larger.

Fig. 4 illustrates the maximum value of j

D

0

(t)jas a function of m

following Theorem 2 when mis dimensionless and

D

0

(t) is ex-

pressed in units of measure used to evaluate project duration in

relation to productivity threshold p

t

. We can assume mP1 since

the levels of initial performance can never exceed the performance

threshold.

Two important observations related to Theorem 2 should be

noted:

1.

D

0

(t) will reach minimum if p

0

remains close to 30% of the per-

formance threshold (mremains close to constant e), which cor-

responds again to situation discussed in relation to Theorem 1.

2. When m> 3 (initial performance is in the lower range) the dif-

ference between the area under the curves will increase with m.

The decision on which curve should be used to track perfor-

mance changes must be made based on other factors such as

team dynamics, amount of consultants, in-house training incor-

porated into the implementation schedule, familiarity with

other team members, consultants, etc.

D

0

(t) will be relatively

larger than if m was close to 3 for both short implementations

(project with an aggressive schedules and implementation

times at 3–4 months) and longer implementations (Fig. 3).

We will discuss the selection of the functional form of the pro-

gress curve for the case, where initial performance is low

(m= 10) in more details in Sections 3 and 4.

We will now deﬁne Theorem 3, to include situations where the

project team exceeded the critical level of performance before the

commencement of the project. The difference between the logistics

and S-curves measured as the difference in areas underneath those

curves at time tscaled by p

t

(Fig. 5) will now be represented as

D

(t)

(Eq. (A.2)

0

).

Theorem 3. When m <e (i.e. p

0

is larger than 30% of performance

threshold), it makes no difference which curve is used to track

performance since both the S-curve and logistic curve become

identical in relatively short time.

The reader should note that for those high levels of initial per-

formance we use the initial condition of P

l

(0) = p

0

to calculate the

difference between the logistics and S-curves, and the second

equation (3.4) represents the logistic curve.

D

(t) is calculated from

Eq. (A.2)

0

.Theorem 3 is depicted in Fig. 6, in which

D

(t) is plotted as

a function of time for various sets parameters: kand m.

Following from Theorem 3 and Fig. 6, the S-curve becomes clo-

ser to the logistic curve if the project team shows higher levels of

initial performance (depicted as plots for m= 1.1 and 1.5). If team’s

initial performance level is very high (m< 2) then the situation on

the ERP project will be similar to the other industrial projects (in a

sense that very little if at all initial training will be provided) and

the progress curve will represent performance increases due to

team cohesion and integrated work patterns.

3.4. Context for applying the models

We will now discuss the context for application of the progress

functions. In each project situation the team has the capacity to

learn and accomplish a ﬁnite amount of work within ﬁxed time

period. We call this the knowledge absorption capacity coefﬁcient

T

0k

. The relationship between planned duration T

0

, the progress

curve coefﬁcient k and the knowledge absorption coefﬁcient is de-

ﬁned in (4.1)

T

0k

¼kT

0

:ð4:1Þ

Combining the progress coefﬁcient and the project duration into a

single variable T

0k

enables critical analysis of the project situation

as Eq. (4.1) also deﬁnes the tradeoff between project duration and

Fig. 4. The maximum value of the difference between the logistic and S-curves as a function of m.

0

Pl, Ps

t

Performance Threshold p t

Progress Function Ps:

S-curve

Progress Function Pl, Pl(0)=p0

Logistic curve

Fig. 5. The difference between the logistic (initial condition of p

0

) and S-curves

measured as the difference in areas underneath those curves at time tfor cases

when m<e.

M. Plaza et al. / European Journal of Operational Research xxx (2009) xxx–xxx 5

ARTICLE IN PRESS

Journal of Operational Research (2009), doi:10.1016/j.ejor.2009.01.010

learning. We can analyze the impact of learning on duration (slower

learning extends the project duration and faster learning shortens

project duration). Further, since kis expressed in [1/‘‘] unit of mea-

sure used for project duration”, T

0k

is dimensionless and can repre-

sent planned project duration, if kremains ﬁxed.

For the purpose of illustrating the application of the models we

deﬁne four categories of IS implementation projects (cf. Table 1).

Implementation projects that ﬁt categories I and II can be modeled

using either the S-curve or logistic curve and to these Theorem 1 is

applicable. When the project team is given intensive initial in-

house training (team bonding/cohesion has occurred) and medium

to high support from external consultants is available, the project

startup effect can be minimized and the logistic curve applicable.

However, if the team members received training at different loca-

tions and team bonding/cohesion was not achieved the startup ef-

fect could be considerable and the S-curve will better represent

performance changes.

Projects in category III are those in which the team has very lit-

tle knowledge or experience in the software they are implementing

and only medium external consulting support is available. The ini-

tial performance of the category III project team is also much less

that 30% of the performance threshold. In such situations Theorem

2is applicable, and we would suggest the logistic equation with

the initial condition P

l

(0) = 0, rather than an S-curve. Fig. 7 below

presents the implications of using the logistic curve in this situa-

tion. We compare the difference between the logistic and S-curves

for different levels of initial team performance (m= 1.5, 3, 6, 10,

and 14). In Fig. 7

D

(t) is scaled by T

0

and plotted as a function of

kT

0

to accommodate the combined effect of project duration and

the progress curve coefﬁcient.

2

In can be seen that when m=3

the plot representing the difference between the two curves re-

mains relatively small for all values of kT

0

. For cases where m<3,

a logistic curve with initial condition of P

l

(0) = 0 will not be used.

If m> 3, the increased values of

D

(t) scaled by T

0

are present only

for very low values of kT

0

(implementation times less than 3

months) and even then they are below the level of 0.4 of the per-

formance threshold.

For implementations projects within Category IV it is assumed

that the project team has a high level of competence obtained from

prior experience implementing the same software elsewhere. Con-

sequently, the team members need no training before the com-

mencement of the project and require minimal consulting

support during the implementation. In this situation Theorem 3

is applicable and either the logistic curve with initial condition

P

l

(0) = p

0

or the S-curve can be used. Fig. 8 below presents the

implications of using the logistic curve in this situation. We com-

pare the difference between the logistic and S-curves for different

levels of initial team performance (m= 1.1, 1.5, 2, 3 and 10). In

Fig. 8

D

(t) is scaled by T

0

and plotted as a function of kT

0

to accom-

modate the combined effect of project duration and the progress

curve coefﬁcient.

Fig. 6. The difference between the logistic and S-curves as a function of time: plotted for various sets of parameters kand m.

Table 1

Four categories of IS projects.

Project categories Level of initial performance p

0

Amount of initial training prior to implementation

a

Amount of consulting support during the implementation

b

I Moderate (mclose to 3) High Very high

II Moderate (mclose to 3) Medium Low

III Very low (m> 4) Low Medium

IV Very high (m< 2) None Very low

a

The following criteria were used for classiﬁcation: ‘‘low” – 2 weeks or less, ‘‘medium” – more than 2 weeks up to 3 weeks, ‘‘high” – more than 3 weeks.

b

The classiﬁcation was based on the ratio of consultants to team members: ‘‘low” – 0.25 or less, ‘‘medium” – more than 0.25 but less than 0.5, ‘‘high” – 0.5 and higher but

less than 1, ‘‘very high” – 1 and higher.

Fig. 7. Comparison of logistic and S-curves for Category III projects.

2

Note that D0ðtÞ

T0in Figs. 5 and 6 is dimensionless, expressed in the units of measure

scaled by performance threshold p

t

.

6M. Plaza et al. / European Journal of Operational Research xxx (2009) xxx–xxx

ARTICLE IN PRESS

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4. The case studies

The data used to illustrate our learning curve analysis was col-

lected from four ERP implementation projects in three companies.

One of the authors was a participant observer on three of the pro-

jects and conducted post-implementation data collection on the

fourth. All three companies are located in Canada. They vary in

size, age and industry. Due to conﬁdential agreements we cannot

disclose the names of the companies in our study; we refer to them

as Company A, B or C. Company A is manufacturer and processor of

vegetable oils and meals. It was established in 1992, has 750

employees and achieves approximately CD$100 million in total

sales, 50% of which is from foreign exports. Company B is a mem-

ber based organization providing services in standards and certiﬁ-

cations. It was established over 80 years ago and has total annual

sales of CD$150 million. Company C is a global manufacturer of

spice ingredients, established in 1919. It is privately owned and

has 2500 employees. It supplies spices to various players (super-

market chains, restaurants, and the food industry). The common

thread in each of these companies is that they all implemented

some type of ERP system and ﬁt the requirements for our analysis.

Table 2 provides a summary of basic data about the implementa-

tion projects. Complete data including project schedules, budgets

and team characteristics and learning activities were collected

for each of Projects 1, 2, 3 and 4.

4.1. Details of the projects

Project 1 concerns the implementation of several modules of the

SAP ERP software over a nine month period. The project team was

composed of 20 employees drawn from the company and 20 exter-

nal consultants. Each team member received 15 days of in-class

training at various SAP training centers on the various modules

that were scheduled to be implemented. They reached a level of

performance of 30% before the commencement of the implementa-

tion (m=3). Following our previous discussion we place this

implementation project in Category I (Table 1). The following

observations were made during the implementation: (1) The ratio

of internal staff to consultants during the project was 1:1. The pro-

ject team had not bonded, so a start up effect was observed during

the early stages of the project. (2) Even though extensive consult-

ing support was provided, the team operated just below its

performance threshold the ﬁrst four months of the project. Using

the S-curve we were able to estimate kclose to 0.8 (see Table 3).

(3) To meet the planned implementation schedule excessive over-

time was required close to the go live week, and additional post

implementation phase work was required. We estimated that

approximately ﬁve additional weeks (

D

Tobserved = 1.25 months),

would have been added to the initial project schedule had the

manager been able accurately predict the performance of the team.

Project 2 concerns the implementation of BAAN ERP software

with a Supply Chain Management extension. The project lasted

one year (T

0

= 12 months). Members of Project Team received in-

house training, during which some level team cohesion was

accomplished. A few members attended advanced level classes at

various locations. The team reached a 30% level of performance be-

fore the commencement of the implementation (m= 3) and fully

bonded, so the start up effect was not observed. We classify this

implementation project as Category II (cf. Table 1). The following

observations were made during the implementation: (1) The con-

sultant to staff ratio during the project was 1:4 ratio. (2) After a lit-

tle over four months, the team was operating close to their

performance threshold. Using the logistic curve we were able to

estimate kas close to 0.8 (cf. Table 3). (3) However, the implemen-

tation schedule was extended by ﬁve additional weeks (

D

Tob-

served = 1.25 months).

Project 3 concerns the ﬁrst implementation of the BAAN ERP

software in one location of Company A. Subsequently, the system

was rolled out to three other factories in Ontario and then to facto-

ries in Alberta and Quebec. Project 3 took one year (T

0

=12

months). A few members received limited training from the soft-

ware provider at various locations. We estimated that the team

reached a 15% level of performance before the commencement of

the project (m= 10). The project team received extensive in-house

training during the early stages of the project and some support

from external consultants. A start up effect was not observed,

and we believe that this was because the team was selected from

members of middle management who already had a very good

Table 2

Comparison of the progress curves for the four projects.

Project and team

characteristics

Project 1

(Company B)

Project 2 (Company C) Project 3 (Company A) Project 4 (Company A)

Project duration [months] 9 12 12 9

Participant observation of

project

7 months 9 months Post implementation data collection 9 months

Modules implemented MM, FI, CO,

Project

Distribution, manufacturing, ﬁnance,

constraint based planning

Distribution, process, manufacturing

excluding MPS and MRP, ﬁnance

Cost accounting, ﬁnance,

distribution, manufacturing,

Team size 20 12 8 8

Team avg. IT experience

(prior to the project)

Over 2 years 1–2 years 1–2 years 3–4 years

Team avg. ERP experience

(prior to the project)

1 month 1 month None 2 years

No. Consultants 20 3 2–4 No full time consultants

Consultants avg. experience 1 year 1–2 years 3 years N/A

No. Team training days (per

member)

15–20 days 15 days 10 days None

Was the implementation

completed

Yes Yes Yes Yes

Fig. 8. Comparison of logistic and S-curves for Category IV projects.

M. Plaza et al. / European Journal of Operational Research xxx (2009) xxx–xxx 7

ARTICLE IN PRESS

Journal of Operational Research (2009), doi:10.1016/j.ejor.2009.01.010

working relationship and were also able to further bond during the

in-house training. We classiﬁed this project as Category III (cf. Ta-

ble 1). During the project consultant to staff ratio remained at 1:4

ratio. Extensive consulting support was given only during the early

stages of the project to reinforce training. By the sixth month of the

project, the team was operating very close to their performance

threshold. Using the logistic curve we were able to estimate kas

close to 0.7 (cf. Table 3). While the implementation schedule was

not extended, excessive overtime was required close to the go-live

week and additional post implementation work was required. We

estimate that had the manger accurately predicted team perfor-

mance approximately six more weeks (

D

Tobserved = 1.5 months),

would have been added to the project schedule.

Project 4 concerns the implementation of a heavily customized

version of the BAAN ERP software in Company A. This project

was completed in nine months (T

0

= 9 months), with the team

reaching a very high level of performance (m= 1.5) before the com-

mencement of the project. This is due to the fact that several of the

team members had participated in the previous implementation.

However, although the majority of the team already worked to-

gether, the start up effect was observed. We believe that this was

due to the fact that no in-house training was provided and team

did not have a chance to bond before starting the project. We clas-

sify this project as Category IV (cf. Table 1). Consulting support was

very minimal and only given on as needed basis to answer team

members’ questions. The team demonstrated a very steep progress

curve and was operating close to their performance threshold dur-

ing the third month of the project. Using the S-curve we were able

to estimate kas close to 0.9 (cf. Table 3). Extension to the imple-

mentation schedule was not required and post implementation

support was estimated as two additional weeks for minor error

correction and data clean up (

D

Tobserved = 0.5 months).

4.2. Discussion of the analysis

The additional time or work required beyond the initial project

duration was registered at the end of each project and then con-

verted to months of extension to the original project duration de-

picted as ‘‘

D

Tobserved” in Table 4. The required extensions were

also calculated from equations presented in Appendix C, which

were derived from the three progress curves. The results are de-

picted in section ‘‘

D

Tcalculated” in columns P

l0

,P

l

and P

s

for two

logistic curves and the S-curve correspondingly. In the construction

of Table 4 we used one month as a duration unit of measure thus,

‘‘

D

Tobserved”, ‘‘

D

Tcalculated” and T

0

are depicted in months. We

present the values of extensions to original project duration (de-

rived from the progress curves) scaled by the values observed on

the projects in section ‘‘

D

Tcalculated/

D

Tobserved”. We also scaled

the total time required to complete the four projects when pro-

gress is tracked by the three progress curves by the total time ob-

served. We present the results in section ‘‘Tcalculated/Tobserved”.

The middle line for each project represents the actual parame-

ters as registered or calculated from Eqs. (C.2–C.5). The top and

bottom lines are presented to show how sensitive the calculations

are in regards misjudgment of k. The columns in grey present the

results of using the ‘‘incorrect” progress curve following the argu-

ment from our previous discussion. We summarized the analysis of

the results depicted in Table 4 into the following observations:

1. For the ﬁrst two projects the results of ‘‘

D

Tcalculated/

D

T

observed” based on S-curve are similar to the results based on

the logistic curve with the initial conditions of P

l

(0) = 0 and very

different from the results based on the logistic curve with the

initial conditions of P

l

(0) = p

0

. The results remain within ±13%

Table 3

Parameters and progress curve coefﬁcients used in our calculations.

Project Parameters Equation for

calculations of

progress curve

coefﬁcient k

Progress

curve

coefﬁcient

k

mN

0

of months

after which

performance

reached

Team

performance

levels

1 3 4 0.97 (3.4.2)

0

0.8

2 3 4.5 0.97 (3.4.1)

0

0.8

3 10 5 0.97 (3.4.1)

0

0.7

4 1.5 2.5 0.97 (3.4.2)

0

0.9

Table 4

Comparison of the progress curves for the four projects.

Project

Parameters

ΔT calculated

ΔT calculated/

ΔT observed

T calculated/

T observed

ΔT

obser-

ved/

T0

ΔT calculated/T0

[%]

PROJECT

T0 k M

Δ

T

observed

Pl0 P

l P

s P

l0 P

l P

s P

l0 P

l P

s [%] Pl0 P

l P

s

0.7 1.42 0.95 1.57 1.14 0.76 1.26 1.02 0.97 1.03 15.8 10.6 17.4

0.8 1.25 0.83 1.38 1.00 0.66 1.10 1.00 0.96 1.01 13.9 9.2 15.3

1

9

0.9

3 1.25

1.1 0.75 1.23 0.88 0.60 0.98 0.99 0.95 1.00

13.9

12.2 8.3 13.7

0.7 1.45 0.94 1.56 1.16 0.75 1.25 1.02 0.98 1.02 12.1 7.8 13.0

0.8 1.22 0.84 1.38 0.98 0.67 1.10 1.00 0.97 1.01 10.2 7.0 11.5

2

12

0.9

3 1.25

1.1 0.74 1.25 0.88 0.59 1.00 0.99 0.96 1.00

10.4

9.2 6.2 10.4

0.6 1.66 1.5 3.85 1.11 1.00 2.57 1.01 1.00 1.17 13.8 12.5 32.1

0.7 1.45 1.28 3.3 0.97 0.85 2.20 1.00 0.98 1.13 12.1 10.7 27.5

3

12

0.8

10 1.5

1.25 1.15 2.85 0.83 0.77

1.90 0.98 0.97 1.10

12.5

10.4 9.6 23.8

0.8 1.25 0.42 0.51 2.50 0.84 1.02 1.08 0.99 1.00 13.9 4.7 5.7

0.9 1.12 0.38 0.46 2.24 0.76 0.92 1.07 0.99 1.00 12.4 4.2 5.1

4

9

0.99

1.5 0.5

1.05 0.35 0.43 2.10 0.70 0.86 1.06 0.98 0.99

5.6

11.7 3.9 4.8

8M. Plaza et al. / European Journal of Operational Research xxx (2009) xxx–xxx

ARTICLE IN PRESS

Journal of Operational Research (2009), doi:10.1016/j.ejor.2009.01.010

tolerance levels if kvaries between [0.7 and 0.9] and are within

[98%, 110%] of the observed extension to the planned project

duration. The observations are in good agreement with and con-

ﬁrm the trend depicted in Theorem 1.

2. For the third project the results of ‘‘

D

Tcalculated/

D

Tobserved”

are very similar for both logistic curves and very different from

the results based on the S-curve. The results remain within

±14% tolerance levels if kvaries between [0.6 and 0.8] and are

within [85%, 97%] of the observed extension to the planned pro-

ject duration. Although the calculated values of extension to ini-

tial durations are below the results observed, the results

presented in Table 4 conﬁrm the trend depicted in Theorem 2.

We would like to point out that the results are very close to

what was actually observed if a logistic curve with the initial

conditions of P

l

(0) = 0 is used for the calculation of

D

T.

3. For the fourth project the results of ‘‘

D

Tcalculated/

D

T

observed” based on S-curve are similar to the results based on

the logistic curve with the initial conditions of P

l

(0) = p

0

and

very different from the results based on the logistic curve with

the initial conditions of P

l

(0) = 0. The results remain within

±16% tolerance levels if kvaries between [0.8 and 0.99] and

are within [76%, 92%] of the observed extension to the planned

project duration. Although the calculated values of extension to

initial durations are again below the results observed, the

results presented in Table 4 conﬁrm the trend depicted in The-

orem 3. We would like to point out that the results are again

very close to what was actually observed if a logistic curve with

the initial conditions of P

l

(0) = p

0

is used for calculation of

D

T,as

predicted in our previous discussion.

5. Conclusions

Researchers have identiﬁed learning and knowledge transfer in

ERP implementation projects as a signiﬁcant problem that requires

attention (Bingi et al., 1999; Markus et al., 2000; Robey et al.,

2002). However, there has been limited research on modeling

and understanding this problem. This research makes four contri-

butions to understanding and managing the learning and knowl-

edge transfer problem in ERP implementation: (1) It presents a

comparative analysis of learning curves and proposes how they

can be used to help ERP implementation planning and manage-

ment. (2) Based on empirical data from 4 ERP implementation pro-

jects, it provides illustrations of how managers can apply the

curves in different project situations. (3) It provides a theoretical

basis for empirical studies of learning and ERP (and other IT)

implementations in different organizational settings. (4) It pro-

vides empirical justiﬁcation for the development of learning curve

theory in IT implementation.

In this paper we presented an analysis of the general logistic

and S-curves and suggest that these curves can be utilized in differ-

ent situations of IT implementation. To illustrate how these curves

can be utilized to improve planning and management of ERP pro-

jects, we presented and analysis using empirical data from four

ERP implementation projects in three companies. Based on theo-

retical explication and our observations, we conclude that a logistic

curve is a good approximation of the performance changes on a

majority of ERP implementations. The logistic curve with the initial

condition of P

l

(0) = 0 can be used on ERP implementation projects

where the levels of initial performance reaches 30% (i.e. m is equal

or greater than 3). The start up effect can be neglected especially if

the team goes through the early stages of team integration during

the in-house training and is guided by either the instructor or a

group of very experienced consultants. The start up effect was,

however, observed in a team that achieved the level of initial per-

formance close to 30% but members received individual training

and therefore integration was not accomplished during training

phase of the project. Although it was demonstrated that both

curves will yield similar results for estimation of total work re-

quired on the project an S-curve should be used in such case.

We also conclude that in cases where team members already

are at a high initial level of performance due to either previous

experience with the system or intensive training, both the logistic

curve with the initial condition of P

l

(0) = p

0

and S-curve can be

effectively used to track performance. However, when the start

up effect is observed (or expected), the S-curve is a better model

than logistic curve for tracking team performance. This observation

can be explained by the effect of team dynamics, which slows

down the performance during the early stages of the project. When

team dynamics are not managed by early ERP training and team

integration strategies high levels of initial performance will not

be achieved and the start up effect will be observed. In this situa-

tion the logistic curve with the initial condition of P

l

(0) = 0 can be

used to track team performance, and is especially effective in situ-

ations where team is at a relatively low initial performance level in

relation to the performance threshold.

As we have shown in our analysis, it is important for ERP imple-

mentation managers to understand the impact of the start up ef-

fect on project duration. Our analysis offers managers direct

insight into both the impact of the start up effect and the impact

of intensive in-house training on project planning and manage-

ment. Our models are easy to implement on ordinary spread pro-

grams and as such are immediately useful to managers. Our

research also provides a basis for empirical studies in ERP imple-

mentation to deﬁne learning curve effects in different implementa-

tion situations. Such studies can help the development of better

management techniques that could lead to improvements in the

cost performance of ERP systems.

Appendix A

In this Appendix we perform a comparative analysis between

the progress functions P

s

and P

l

used in (3.2) and (3.4) respectively.

In particular, we compare the areas underneath the two curves for

tP0 and discuss the difference for large times. Deﬁning

D

(t)as

the difference in area underneath the two curves at time tscaled

by p

t

, we have

D

ðtÞ 1

p

t

Z

t

0

½P

l

ðsÞP

s

ðsÞds

¼1

p

t

Z

t

0

p

t

ð1e

ks

Þ p

t

1þAe

ks

ds:ðA:1Þ

Evaluation of the integral and rearranging gives

D

0

ðtÞ¼

1

k

½lnðAþ1Þ1

þ

1

k

e

kt

1

k

lnðAe

kt

þ1Þ

D

ðtÞ¼

1

k

½lnðAþ1Þ

m1

m

þ

1

k

m1

m

e

kt

1

k

lnðAe

kt

þ1Þ

8

>

>

>

>

<

>

>

>

>

:

if P

l

ð0Þ¼0;

P

l

ð0Þ¼p

0

:ðA:2Þ

Since according to (3.2) A=m1, (A.2) can be rewritten into (A.2

0

)

D

0

ðtÞ¼

1

k

½ln m1þ

1

k

e

kt

1

k

ln½ðm1Þe

kt

þ1;

D

ðtÞ¼

1

k

½ln m

m1

m

þ

1

k

m1

m

e

kt

1

k

ln½ðm1Þe

kt

þ1:

(ðA:2

0

Þ

From this we see that as t?1,

D

has a well-deﬁned limit, and the

value of this asymptote is

Llim

t!1

D

ðtÞ¼

1

k

½ln m1¼

1

k

ln

p

t

p

0

1

hi

1

k

ln m

m1

m

¼

1

k

ln

p

t

p

0

p

t

p

0

p

t

hi

8

>

<

>

:

if P

l

ð0Þ¼0;

P

l

ð0Þ¼p

0

:

ðA:3Þ

M. Plaza et al. / European Journal of Operational Research xxx (2009) xxx–xxx 9

ARTICLE IN PRESS

Journal of Operational Research (2009), doi:10.1016/j.ejor.2009.01.010

The sign of this limit will depend on the ratio of the productivity

threshold to the initial productivity. In cases when we compare

the logistic curve under the initial condition of P

l

(0) = 0 (condition

most commonly used to represent the situation when initial level

of performance is low, or p

0

60.33*p

t

) to the S-curve, we see that

if the productivity threshold is exactly ep

0

, then both curves become

identical in a relatively short time and it makes no difference which

curve is used for analysis, as depicted in (A.4).

L>0ifp

t

>ep

0

;

L¼0ifp

t

¼ep

0

;and

L<0ifp

t

<ep

0

:

8

>

<

>

:

ðA:4Þ

Appendix B

Here we compute the maximum value of j

D

(t)jwhere

D

(t) is the

difference in the area underneath the two curves at time tscaled

by p

t

, whose form is given in (A.2). We will focus again on the sit-

uation when the initial level of performance is low, which is repre-

sented by the logistic curve under the initial condition of P

l

(0) = 0

(ﬁrst equation in (A.2)).

Since

D

(t) is a continuous function,

D

(0) = 0 and Llim

t!1

D

ðtÞ

is positive, negative or zero depending on the value of

p

t

p

0

(recall

(A.4)) but ﬁnite, it follows that the maximum value of j

D

(t)jwill

occur either at a critical point for

D

(t)orast?1. Differentiating

D

(t) gives

D

0

ðtÞ¼ e

kt

Ae

kt

þ1ðAAe

kt

1ÞðB:1Þ

from which it follows that

D

0

(t) < 0 for all time if p

t

62p

0

, and has a

local minimum (found by setting the derivative equal to zero and

solving for t)at

t

¼1

kln p

t

2p

0

p

t

p

0

;ðB:2Þ

if and only if p

t

>2p

0

. Note that we used the deﬁnition for Agiven

(3.2) to rewrite this result.

It follows that the maximum value of j

D

(t)jis given by

M¼max j

D

ðtÞj ¼ jLjif p

0

P

p

t

2

;

maxfj

D

ðt

Þj;jLjg if p

0

<

p

t

2

:

ðB:3Þ

Here t

*

is given in (B.2),

D

(t) is given in (A.2) and Lis given in (A.4).

Appendix C

In order for the total work required for system implementation

to be the same if the progress curve is represented by either of the

progress functions: P

s

(Eq. (3.2),P

l0

(1st equation from (3.4)), P

l

(2nd equation from (3.4)) or constant performance function p

t

,

we require

Z

T

0

0

p

t

dt ¼R

T

0

p

t

ð1e

kt

Þdt

R

T

0

p

t

1

m1

m

e

kt

dt

R

T

0

p

t

1þðm1Þe

kt

dt

8

>

>

<

>

>

:

where

P

l0

ð0Þ¼0;

P

l

ð0Þ¼p

0

;

P

s

ð0Þ¼p

0

;

ðC:1Þ

which can be integrated and divided by p

t

to give

kT þe

kt

¼kT

0

þ1 if logistic curve and P

l0

ð0Þ¼0;ðC:2Þ

kT þm1

me

kt

¼kT

0

þm1

mif logistic curve and P

l

ð0Þ¼p

0

;

ðC:3Þ

kT þlnððm1Þe

kt

þ1Þ¼kT

0

þln mif S-curve:ðC:4Þ

We solved equations (C.2–C4) numerically for the given choice of

parameter values depicted in Table 2. The required extension to

the original project duration

D

Tcan be then calculated from Eq.

(C.5), where Tis the numerical solution of either (C.2), (C.3), or

(C.4) depending on whether one of the logistic curves or S-curve

represents the progress function and T

0

is the planned project dura-

tion calculated from the constant performance function p

t

.

D

T¼TT

0

:ðC:5Þ

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