An alternative for robust estimation in Project Management

European Journal of Operational Research 220 (2012) 443–451

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Stochastics and Statistics

An alternative for robust estimation in Project Management M.M. López Martín a, C.B. García García a,⇑, J. García Pérez b, M.A. Sánchez Granero c a

Department of Quantitative Methods for Economics and Business, University of Granada, Campus de Cartuja, s/n, Granada 18071, Spain Department of Applied Economics, University of Almería, Ctra. La Cañada de San Urbano, s/n, Almería 04120, Spain c Department of Geometry and Topology, University of Almería, Ctra. La Cañada de San Urbano, s/n, Almería 04120, Spain b

a r t i c l e

i n f o

Article history: Received 23 February 2011 Accepted 31 January 2012 Available online 6 February 2012 Keywords: Project Management Uncertainty modeling Distribution Simulation

a b s t r a c t Recently, Hahn (2008) has proposed the mixture between the uniform and the beta distributions as an alternative to the beta distribution in PERT methodology which allows for varying amounts of dispersion and a greater likelihood of more extreme tail-area events. However, this mixture lacks a closed cumulative distribution function expression and its parameters remain a difficult interpretation. In addition, the kurtosis limit of the beta distribution is 3. Due to their higher kurtosis and easier elicitation we consider the Two-Sided Power and the Generalized Biparabolic distributions good candidates for applying Hahn’s idea (2008) and for building the Uniform-Two Sided Power (U-TSP) and the Uniform-Generalized Biparabolic (U-GBP) distributions. Using the same example from Hahn (2008) we are going to demonstrate that we can obtain more accuracy and a greater likelihood of more extreme tail area events. These distributions could be applied in other heavy-tailed phenomena which are usual in business contexts. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction The need for a more flexible distribution which allows a greater likelihood of more extreme tail-area events and different amounts of dispersion has been highlighted recently by Hahn (2008). This issue had already been raised in different fields: Moore (1964) and Fama (1965) noted that the empirical distribution of financial return was leptokurtic and with tails thicker than normal. Embrechts et al. (1997), Grant et al. (2006) and Juneja et al. (2006) also presented financial applications that involved variables with heavy tails. Juneja (2007) used variables with heavy tails to estimate the probability of a long delay in the completion of an activity. TzungCheng et al. (2008), building on Beaman (2006), demonstrated the need for robust estimators in distributions with long tails and warned of the dangers of ignoring the existence of outliers, even in these cases. Hahn (2008) (relying primarily on McCullagh and Nelder, 1989; Mitchell and Zmud, 1999; Steele and Huber, 2004; Grant et al., 2006; Atkinson et al., 2006) raised the same question establishing the need to develop robust estimation methods in the PERT methodology and proposing a mixture distribution which allows greater flexibility in fitting the data (Gelman et al., 2003). As is well-known, the beta distribution (1) has been the traditional underlying distribution of this methodology (see Yu Chuen-Tao, 1980, 1989) and its probability density function (pdf) is given by ⇑ Corresponding author. Tel.: +34 958 248 344. E-mail addresses: [email protected] (M.M. López Martín), cbgarcia@ ugr.es (C.B. García García), [email protected] (J. García Pérez), [email protected] (M.A. Sánchez Granero). 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2012.01.058

f ðxja; bÞ ¼

Cða þ bÞ ðx  aÞa1 ðb  xÞb1 ; CðaÞCðbÞ ðb  aÞaþb1

ð1Þ

where a < x < b, a > 0, b > 0. Malcolm et al. (1959) suggested the following expression for the mean and the variance of the activity time distribution:

E½X ¼

a þ 4m þ b ; 6

ð2Þ

ðb  aÞ2 ; 36

ð3Þ

and

var½X ¼

where a, m and b are the lower bound, the most likely and the upper bound, respectively. These expressions prescribe a light tailed distribution which does not consider sufficiently the key factor of the tail area of an activity time. Based on this, Hahn (2008) presented a mixture between the uniform and the beta distributions which permits a greater likelihood of more extreme tail-area events. We can find some precedents in the work of Sungyeol and Serfozo (1999), van Dorp et al. (2006, 2007) and Mehrotra et al. (1996) where different mixture distributions are satisfactorily applied. In addition to the need to introduce a robust estimation in the field of Project Management, Hahn (2008) insisted on the criticism of the PERT expressions based on the fact that (2) and (3) cannot be obtained from (1) (Sasieni, 1986; Gallagher, 1987; Littlefield and Randolph, 1987). According to Golenko-Ginzburg (1988), the mean and variance expressions corresponding to (1) can be expressed by

444

E½X ¼

M.M. López Martín et al. / European Journal of Operational Research 220 (2012) 443–451

a þ km þ b ; kþ2

(

ð4Þ

bm

and 2

var½X ¼

 xa 2n

g ðx=a; m; b; nÞ ¼ C ðnÞ ma2n bx

k ðm  aÞðb  mÞ þ ðk þ 1Þðb  aÞ2 ðk þ 3Þðk þ 2Þ2

ð5Þ

:

This fact, together with the problem of obtaining the four parameters of (1) from the three values provided by an expert was solved by Herrerías et al. (2003), in line with Sasieni (1986) and Kamburowski (1997). By establishing the beta distribution as being similar to the Normal distribution, i.e. mesokurtic and with constant vari1 ance ðr2 ¼ 36 Þ, Herrerías et al. (2003) obtain the classical PERT expressions (2) and (3) from the general beta distribution (1), (Sculli, 1989). This distribution, obtained when k ¼ a þ b  2 ¼ 4 in expressions (4) and (5), was called Type I PERT beta distribution by Hahn (2008) and we shall call it classical beta distribution. In spite of its flexibility and popularity, the beta distribution lacks a closed cumulative distribution function (cdf) expression and its parameters continue to have a difficult interpretation (Clark, 1962; Grubbs, 1962; Moder and Rodgers, 1968; Moitra, 1990; Kamburowski, 1997; Johnson, 1997; Lau et al., 1997). Therefore, although the mixture proposed by Hahn (2008) is more flexible and more tailored to the situation than the beta distribution, it still presents the disadvantage of no closed formula for the cumulative distribution function (cdf), which will be desirable for the development of the elicitation for the parameters of the distribution. Van Dorp and Kotz (2002a) presented the Two-Sided Power (TSP) distribution as an alternative, more peaked and with a closed expression for cdf, to the beta distribution. Its probability density function is given by

n f ðx=a; m; b; nÞ ¼ ba

(

 xa n1 ; ma  bx n1 ; bm

if a < x 6 m;

ð6Þ

if m < x < b:

García et al. (2005) obtained the classical TSP distribution, when n = 3.02344, being mesokurtic and having constant variance and compared it to the classical beta distribution applied in the PERT methodology. In a similar form to the TSP distribution, the Generalized Biparabolic (GBP) distribution (García et al., 2009) has a closed form for the cumulative distribution function (cdf) too, but, contrarily to the TSP distribution, its density function is smooth at the mode. This distribution has been admitted as a good alternative underlying distribution in the PERT methodology (García et al., 2010). Its pdf is given by

A

 xa n

2

if a < x 6 m;

ma

;

bm

; if m < x < b;

 bx n

ð7Þ

ð2nþ1Þðnþ1Þ where CðnÞ ¼ ð3n1ÞðbaÞ . It can be proved that the classical GBP distribution can be obtained when n = 2.74669. By comparing the kurtosis coefficient (b2) of both distributions in Fig. 1A it is noted that the kurtosis coefficient of the classical beta distribution has a longer range than the one for the classical TSP and GBP distributions, but is only higher in the extreme values. In addition, when the expert estimates the maximum, minimum and most likely value, the latter is normally centred on the interval (a, b), where the classical TSP and GBP distributions present a kurtosis greater than the classical beta distribution. In particular, we can see (Fig. 1B) that the symmetrical TSP and GBP distributions always have a kurtosis greater than the beta distribution and, while the limit of the kurtosis of the TSP is the kurtosis of the Laplace (b2 = 6), the limit for the beta distribution is the kurtosis of the normal one (b2 = 3). Note that is the standardized mode. The goal, according to Hahn (2008), is to get a more flexible distribution which allows for varying amounts of dispersion and a greater likelihood of more extreme tail-area events and to be easily elicitable. It is evident that the kurtosis coefficient is a critical factor in this issue. Moreover, the TSP and GBP distributions can be directly elicitated by using an upper quantile, a lower quantile, the most likely value and a third quantile (Kotz and van Dorp, 2006). Therefore, we consider the TSP and GBP distributions better candidates than the classical beta distribution for the mixture proposed by Hahn. The paper is organized as follows. Section 2 introduces the uniform and Two-Sided Power mixture (U-TSP) and the uniform and Generalized Biparabolic mixture (U-GBP) distributions. Their probability density and cumulative distribution functions and their stochastic characteristics are also presented in that section. The elicitation process of the U-TSP and U-GBP mixture distributions are analyzed in Section 3. The same empirical application presented by Hahn (2008), and originally by Moder et al. (1983), has been rebuilt in Section 4 using the U-TSP and the U-GBP distributions. Some concluding remarks are provided in Section 5.

2. Mixture distributions Van Dorp and Kotz (2003) presented a mechanism of generating two-sided continuous distributions with bounded support using the following density:

B

4.5

2

7 6 5

Kurtosis

Kurtosis

4

3.5

3

4 3 2 1

2.5

0 0

10

20

30

40

50

60

70

80

90

100

2 0

0.1

0.2

0.3

Classical Beta

0.4

0.5

0.6

Classical TSP

0.7

0.8

0.9

Classical GBP

1

θ

Symmetrical BETA

Symmetrical TSP

Normal

Laplace

n

Symmetrical GBP

Fig. 1. (A) Comparison between the kurtosis coefficient of the classical beta (k = 4), classical TSP (n = 3.02344) and classical GBP (n = 2.74669) distributions. (B) Comparison between the kurtosis coefficient of the symmetrical beta, symmetrical TSP, symmetrical GBP, normal and symmetrical Laplace distributions.

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(   p ht =w ; if 0 < t 6 h; g ðtjh; pð=wÞÞ ¼  1t  =w ; if h < t < 1; 1h

Therefore, the lower the parameter k the greater the presence of the uniform distribution in the mixture. The uniform distribution is clearly recovered when k ¼ 0 and the Two-Sided Power and the Generalized Biparabolic distributions are recovered in expressions (14) and (15), respectively, when k ¼ 1. Fig. 2 compares the probability density function of the classical U-TSP and the classical U-GBP with the classical U-BETA distributions presented by Hahn (2008). Note that the classical U-TSP distribution is more peaked than the classical U-BETA distribution which is coherent with what was presented in Fig. 1. Note that in both cases the most likely value density for the UBETA distribution is higher than the one for the U-GBP distribution and lower than the one for the U-TSP distribution.

ð8Þ

where p( |w) is called generating density and is an appropriately selected continuous pdf defined on [0,1] with parameter(s) w, which may be a vector value, and h is called the reflection parameter. The cdf associated to (8) is:

( Gðtjh; Pð=WÞÞ ¼

  hP ht =W ; 1  ð1  hÞP

if 0 < t 6 h;  = W ; if h < t < 1; 1h

 1t

ð9Þ

where Pð=WÞ is the cdf of the generating density pð=wÞ. From (9) the following quantile function is obtained:

  G z=h; P1 ð=WÞ ¼ 1

(

  hP1 hz =W ; 1  ð1 



1z = hÞP1 1h



if 0 < z 6 h; 2.1. The U-TSP mixture distribution

W ; if h < z < 1: ð10Þ

By using the generating density (14) and the mechanism of generating two-sided continuous distributions (8) we can obtain the Uniform-Two Sided Power (U-TSP) distribution with the density:

The expression (11), presented by van Dorp and Kotz (2002a,b), relates the moments of the generating variable Y  p( |w) with those of the variable T  g(t|h, p( /w)):

0 1 k k X B C i iþ1 k k i kþ1 E½T  ¼ h E½Y jW þ @ i Að1Þ ð1  hÞ E½Y jW:

( g ðt=pðy=n; k; hÞÞ ¼

ð11Þ

i 8 h  n1 > t k ht þ ð1  kÞ ; > > > > < if 0 < t 6 h; h i Gðt=P ðy=n; k; hÞÞ ¼ > 1  ð1  tÞ k 1t n1 þ ð1  kÞ ; > > 1h > > : if h < t < 1:

It can be proved that the generating density function of the TSP and GBP distributions are given, respectively, by

pðyjnÞ ¼

ð12Þ

 ð2n þ 1Þðn þ 1Þ  2n y  2yn ; 3n  1

ð13Þ

E½T ¼

ð18Þ

" # 1 2kðn2  1Þð2 þ nAÞ  3k2 B2 ðn  1Þ2 ðn þ 2Þk2 1þ ; 12 ðn þ 1Þ2 ðn þ 2Þ ð19Þ

This procedure can be taken as a particular case of the family of Elevated Two Sided Power Distributions (ETSP) introduced by García et al. (2011). Similarly, the Uniform-Generalized Biparabolic (U-GBP) mixture distribution can be obtained from the following generating density function:

A

1 þ n þ k  2hk  nk þ 2hnk ; 2ðn þ 1Þ

var½T ¼

ð14Þ

 ð2n þ 1Þðn þ 1Þ  2n pðyjn; kÞ ¼ k y  2yn þ ð1  kÞ: 3n  1

ð17Þ

The main characteristics of the U-TSP distribution follow immediately using (11) and the relationship between the moments around zero of and (van Dorp and Kotz, 2003):

where n is the shape parameter of the distribution (van Dorp and Kotz, 2002a; García et al., 2009). Since none of the distributions commonly used in the PERT have such heavy tails as the uniform one, we propose adding a mixing parameter in order to get the Uniform-Two Sided Power (U-TSP) mixture distribution. In this case, the generating density is given by

pðyjn; kÞ ¼ knyn1 þ ð1  kÞ:

where A = 6h2  6h + 1 and B = 1  2h. The expression (19) changes to the following one when n ¼ 3:02344:

var½T  ¼

ð15Þ

B

1.6 1.4

i 1 h 1 þ 3:63ð0:94 þ hÞð0:06 þ hÞk  3:03ð0:5 þ hÞ2 k2 : 12 ð20Þ

3 2.5

1.2 2

pd f

pdf

1 0.8 0.6

1.5 1

0.4 0.5

0.2 0

0 0

0.2

0.4 U-BETA

0.6 U-TSP

0.8 U-GBP

1

ð16Þ

The cdf associated with the density (16) is given by

i¼0

pðyjnÞ ¼ nyn1 ;

 n1 þ ð1  kÞ; if 0 < t 6 h; kn ht  1t n1 kn 1h þ ð1  kÞ; if h < t < 1:

t

0

0.2

0.4 U-BETA

0.6 U-TSP

0.8

1

t

U-GBP

Fig. 2. Probability density function of the classical U-BETA, the classical U-TSP and the classical U-GBP mixtures distribution. (A) k = 0.25, (B) k = 0.75.

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Note, Fig. 3A, that the U-BETA distribution always presents a higher variance except when h e [(0, 0.25) [ (0, 0.75)] and k > 0:5. The kurtosis coefficient is always higher in the U-TSP distribution when approximately 0.25 < h < 0.75 for every value of k. Moreover, if k is less than 0.25 the kurtosis coefficient of the U-TSP distribution is higher for every value of h. 2.2. The U-GBP mixture distribution Similarly, by using the mechanism of generating two-sided continuous distributions (8) and the generating density (15), the Uniform-Generalized Biparabolic (U-GBP) density is obtained as:

8 h   n i 2n > < kC ðnÞ ht  2 ht þ ð1  kÞ; if 0 < t 6 h; h  i g ðt=pðy=n; k; hÞÞ ¼   2n n > kC ðnÞ 1t  2 1t : þ ð1  kÞ; if h < t < 1; 1h 1h ð21Þ where CðnÞ ¼ ð2nþ1Þðnþ1Þ . The cdf associated with the density (21) is ð3n1Þ given by 8  2nþ1  nþ1 2ðht Þ > ðht Þ > > if < 0 < t 6 h; < kC ðnÞ ð2nþ1Þ  ðnþ1Þ h þ ð1  kÞt; GðtPðyjn;k;hÞÞ ¼   nþ1 1t 2nþ1 1t > 2ð Þ ð Þ > >  ð1h þ ð1  kÞt: if h < t < 1: : 1  kð1  hÞC ðnÞ 1h ð2nþ1Þ nþ1Þ ð22Þ

The main characteristics of the U-GBP distribution follow immediately using (11) and the relationship between the moments around zero of Y and T (van Dorp and Kotz, 2003):

E½T ¼

2 þ 7n þ 3n2 ð1  kÞ þ 6hn2 k ; 2ðn þ 2Þð3n þ 1Þ

var½T ¼

ð23Þ

36 þ 288n  An2  Bn3  Cn4  Dn5  En6 12ðn þ 2Þ2 ðn þ 3Þð2n þ 3Þð3n þ 1Þ3

ð24Þ

;

where

  A ¼ 809 þ 68 þ 120h  120h2 k;   B ¼ 983 þ 242 þ 732h  732h2 k;     C ¼ 581 þ 92 þ 1416h  1416h2 k þ 243  972h þ 972h2 k2 ;     D ¼ 165 þ 780 þ 972h  972h2 k þ 243  972h þ 972h2 k2 ;     E ¼ 18 þ 36 þ 216h  216h2 k þ 54  972h þ 972h2 k2 : In the particular case when n = 2.74699, the expression (24) becomes:

  var½T ¼ 0:266286 0:3129 þ Ak  Bk2 ; 2

where A = 0.05629  1.1378h + 1.1378h and B = 0.25  h + h . In a similar form to the U-TSP distribution, the U-GBP distribution always presents a lower variance except when h e [(0, 0.25) [ (0, 0.75)] and k > 0:5, see Fig. 4A. The kurtosis coefficient is always higher in the U-GBP distribution when approximately 0.25 < h < 0.75, for every value of k. Moreover, if k is less than 0.25 the kurtosis coefficient of the U-GBP distribution is higher for every value of h. To conclude, Fig. 5 presents the variance of the classical symmetrical U-TSP, U-GBP and U-BETA mixture distributions. The behavior of the variance in the U-TSP is quite similar to the UGBP distribution, but both have less variance than the U-BETA distribution. 3. Elicitation of mixture distributions From the beginnings of the PERT methodology, the use of quantiles has been regarded as an adequate alternative to the direct estimate of the parameters, and (Hampton et al., 1973; Chesley, 1975; Spetzler and Stael Von Holstein, 1975; Wallsten and Budescu, 1983). This elicitation procedure has been supported in different fields by many authors such as Moskowitz and Bullers (1979), Selvidge (1980), Davidson and Cooper (1980), Lau and Somarajan (1995), Johnson (1998), Lau et al. (1997) More recently, Kotz and van Dorp (2006) developed a procedure to elicitate the TSP distribution using quantiles and providing reasons for its use in order to simplify the work of experts and van Dorp et al. (2007) developed a procedure for estimating a trapezoidal distribution using quantiles. 3.1. Elicitation of the Uniform-Two Sided Power mixture distribution In the case of the U-TSP mixture distribution, as with the UBETA, we need the expert to provide k in addition to a, m and b. The interpretation of this mixing parameter is not at all intuitive. Hahn (2008) presents two relatively direct methods to elicitate k in the classical U-BETA distribution that could be applied in the U-TSP. However, the latter has an explicit closed cdf expression which makes the elicitation procedure easier. We propose elicitation of the classical U-TSP (where n ¼ 3:02344), García et al. (2005). In this case, with a, m and b known, the only parameter to be determined is k. The elicitation of this parameter is carried out using only one percentile. First, in order to show that the lower or upper tail of the distribution can be used instead of a central quantile we develop the following proposition from expression (17):

B

0.08 0.06 0.04 0.021 0.75

0 0.25 0.5 0.5

0.75 0.25 0 1

Kurtosis

A

Variance

ð25Þ

2

4 3 2

1 0.75 0.5

0 0.25

0.25

0.5 0.75 1

0

Fig. 3. (A) Variance of the classical U-TSP distribution and the classical U-BETA distribution (squared). (B) Kurtosis coefficient of the classical U-TSP distribution and the classical U-BETA distribution (squared).

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M.M. López Martín et al. / European Journal of Operational Research 220 (2012) 443–451

B

0.08 0.06 0.04 0.021 0.75

0 0.25

4 3 2

Kurtosis

Variance

A

0.5 0.5

1 0.75 0.5

0

0.75

0.25

0.25

0.25

0.5 0 1

0.75 1

0

Fig. 4. (A) Variance of the classical U-GBP distribution and the classical U-BETA distribution (squared). (B) Kurtosis coefficient of the classical U-GBP distribution and the classical U-BETA distribution (squared).

0.09

Similarly, it could be proved that aq 6 q. h Finally, from (17) we can elicitate the parameter k:

0.08

8 p 1ap > > p 6 ap 6 h; > a p < 1ð h Þn1 ; k¼ 1q 11a > q > > : 1aq n1 ; q 6 aq 6 q:

0.07

Variance

0.06 0.05

1

0.04 0.03 0.02 0.01 0 0

0.2

0.4

Classical U-BETA

0.6

0.8

Classical U-TSP

1

Classical U-GBP

Fig. 5. Variance of the classical U-BETA (k ¼ 4; h ¼ 0:5), the classical U-TSP (n = 3.02344; h = 0.5) and the classical U-GBP (n = 2.74669; h = 0.5) distributions.

Proposition. Given two quantiles ap and aq, where ap < h < aq, the following relation is verified:

0 6 Gðap Þ ¼ p 6 ap 6 h 6 aq 6 Gðaq Þ ¼ q 6 1:

ð26Þ

Proof. By substituting in (26):

"  #   n1  n1 ap h p ¼ ap k þ ð1  kÞ 6 ap k þ ð1  kÞ ¼ ap : h h

ð27Þ

A

4

1h

Therefore, if we have a percentile to the left or right of h, by satisfying the conditions stated in (26), we can obtain from any of the terms contained in (28). Indeed, we are not forced to use the classical U-TSP and, what is more, by increasing the value of n we can get distributions with higher kurtosis and a more accurate fit. In this case, we can directly elicitate the parameters k and n of a generic U-TSP distribution (n undetermined) by using two percentiles and solving the system formed by (28) and (29): 1q 1  app 1  1a q  n1 : ap n1 ¼ 1aq 1 h 1  1h

ð29Þ

One must unfortunately note that this system does not always have a solution, but when it is possible to solve it, it allows us to obtain an elicitation of the underlying distribution from percentiles. In the literature on PERT methodology, this procedure is considered closer to reality, (Moskowitz and Bullers, 1979; Selvidge, 1980; Davidson and Cooper, 1980; Alpert and Raiffa, 1982; Keefer and Verdini, 1993; Lau and Somarajan, 1995; Lau et al., 1996; Lau et al., 1997; Johnson, 1997; Kotz and van Dorp, 2006; van Dorp et al., 2007). It can be said that, in the case of U-TSP, we have

B

5

ð28Þ

4

3

3

λ 2

λ 2

1

1 0

0 0.9

1.4

1.9

2.4

2.9

3.4

3.9

0

0.5

1

1.5

n branch 1

branch 2

2

2.5

n branch 1

branch 2

Fig. 6. Graphical solution of an elicitation example. (A) U-TSP distribution. (B) U-GBP distribution.

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M.M. López Martín et al. / European Journal of Operational Research 220 (2012) 443–451

Table 1 Stochastic characteristics of the total project time variable for the classical U-TSP, U-GBP and U-BETA distributions obtained by Monte Carlo simulations.

Classical Classical Classical Classical Classical Classical Classical Classical Classical Classical Classical Classical

U-BETA U-TSP U-GBP U-BETA U-TSP U-GBP U-BETA U-TSP U-GBP U-BETA U-P U-GBP

(k = 0.25) (k = 0.25) (k = 0.25) (k = 0.5) (k = 0.5) (k = 0.5) (k = 0.75) (k = 0.75) (k = 0.75) (k = 1) (k = 1) (k = 1)

Mean

Standard deviation

95% C.I.

Sample kurtosis

51.46 51.77 51.75 50.21 50.83 50.82 48.93 49.87 49.81 47.65 48.91 48.79

5.52 5.4 5.38 5.13 4.96 4.91 4.46 4.32 4.22 3.41 3.4 3.21

(42.20, 62.15) (42.63, 62.17) (42.62, 62.15) (42.04, 61.08) (42.84, 61.21) (42.89, 61.18) (41.95, 59.43) (43.16, 59.75) (43.24, 59.63) (42.01, 55.10) (43.56, 56.63) (43.64, 55.89)

2.24 2.25 2.27 2.54 2.5 2.54 3.14 2.95 3.06 2.93 3.04 2.97

Fig. 7. (A) Distribution of the total project time using the Classical U-BETA mixture distribution: (a) k = 0.25, (b) k = 0.5, (c) k = 0.75, (d) k = 1. (B) Distribution of the total project time using the Classical U-TSP mixture distribution: (a) k = 0.25, (b) k = 0.5, (c) k = 0.75, (d) k = 1. (C) Distribution of the total project time using the Classical U-GBP mixture distribution: (a) k = 0.25, (b) k = 0.5, (c) k = 0.75, (d) k = 1.

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improved the direct elicitation, since from two quantiles it is possible to obtain the parameter k, which measures the degree of mixture, (the greater k the higher the purity of the TSP) and the parameter n, which is associated with the concentration and kurtosis of the distribution (Kotz and Seier, 2007). Alternatively, it is easy to elicitate n using (30) if p 6 ap 6 h or (31) if h 6 aq 6 q, when you have already got k from one of the methods proposed by Hahn (2008):

 1ap Ln 1  k p n¼1þ

ð30Þ

;

a

Lnð hp Þ

and

 1q 11a Ln 1  k q n¼1þ

1a

Lnð 1hq Þ

ð31Þ

:

without the presence of the uniform distribution. The difference between these distributions is small and it increases as k decreases. Note that the empirical distribution (obtained by simulation) has a higher kurtosis coefficient when k = 0.25 for the U-GBP distribution and k = 1 for the U-TSP distribution. For k = 0.5, the U-GBP and UBETA distributions have the same kurtosis. If k = 0.75, the classical U-BETA distribution has the highest kurtosis. In Table 2, we can see that the value of x (time to complete the project) which makes PðX 6 xÞ ¼ 0:95 is shorter when we are working with the classical U-TSP or U-GBP distributions than when we are working with the classical U-BETA distribution with any kind of mixture. Note that the minimum and maximum values of x, written in bold in Table 2, are always correspond to the U-TSP and U-GBP distributions. However, when n varies, the time to complete the project becomes shorter using the U-TSP or the U-GBP distribution. For instance, when k = 1 the time to complete the project with n = 10 is 49.21, which is shorter than the 53.86 obtained from the U-BETA distribution. The explanation of this fact is shown

1q Eqs. (30) and (31) only have solution if k > 1  app and k > 1  1a , q respectively.

Table 2 Value of x which makes PðX 6 xÞ ¼ 0:95 for k = 0.25, 0.5, 0.75 and 1, for classical UBETA, classical U-TSP and classical U-GBP distributions obtained by Monte Carlo simulations.

3.2. Elicitation of the Uniform-Generalized Biparabolic mixture distribution In the case of the U-GBP distribution, as with the U-TSP one, the procedure introduced by Van Dorp et al. (2006) is used. It can be shown that proposition (26) is also verified in this case. From (22) we can elicitate the parameter k:

k¼

8 ðap pÞð1þ3nÞ > > < ap ðap Þn 1 ðap Þn ð1þnÞ13n ; h

> > :

p 6 ap 6 h;

h

a q ð1þ3nÞ

1a ðnq Þ a n  ; h 6 aq 6 q: ðaq 1Þ 1hq 1 ð hq Þ ð1þnÞ13n

ð32Þ

U-BETA

A = 0.25 A = 0.5 A = 0.75 k=1

U-TSF

60.88 59.67 57.66 53.86

U-GBP

n=5

n = 10

n = 3.02344

n=5

n = 10

n = 2.74669

60.9 59.57 57.46 52.35

60.82 59.44 57.15 49.21

60.97 59.94 58.22 55.31

60.87 59.58 57.36 51.91

60.77 59.46 57.24 49.22

60.96 59.87 58.04 54.7

The minimum and maximum values of x for every value of are shown in bold.

As an example, we consider the quantiles (p; ap) = (0.001; 0.003) and (q; aq) = (0.9997; 0.999) and h = 0.4. In the case of the U-GBP distribution, by substituting the value of ap and aq in (32) we shall obtain a function depending on n. The analytical solution is quite cumbersome. We have therefore obtained the solution in a graphical way. The solution is n = 0.6115 and k ¼ 0:7225. By using the same example, we can obtain the solution in the case of the U-TSP distribution only by substituting the values of ap and aq in (28). In this case the solution becomes and n = 1.5006 and k ¼ 0:7297. Fig. 6 shows both graphical solutions. 4. Empirical application We now analyze the performance of the U-TSP and the U-GBP distributions by using the example used by Hahn (2008) and comparing the results. This application, originally from Moder et al. (1983), consists of 29 activities in a real-world electronic module development project having multiple paths. Monte Carlo simulations have been applied to obtain the distribution of the total project time for the classical U-TSP, U-GBP and U-BETA distributions using the same values of k applied by Hahn (2008) (note that the mixture parameter was originally called h). The stochastic characteristics are provided in Table 1. The results obtained using the classical U-TSP and U-GBP distributions are quite similar to the classical U-BETA results. Note that the estimation of the mean is always higher using the classical U-TSP and U-GBP distributions. The intervals are slightly deviated to the right in the case of the U-TSP distribution. In both cases, increasing the presence of the uniform distribution makes the intervals greater. Plots of the distribution of the total project time appear in Fig. 7. We see that the variance is always higher using the classical UBETA distribution than using the classical U-TSP and U-GBP, even

Fig. 8. (A) Representation of the classical U-BETA, classical U-TSP and classical UGBP cumulative distribution function for k = 0.75. (B) Representation of the U-BETA, U-TSP and U-GBP cumulative distribution function for k = 0.75 and different values of n.

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Fig. 9. The 95% percentile of the time needed to complete the project by using the classical U-BETA, U-TSP and U-GBP distributions. (A) k = 0.25; (B) k = 0.5; (C) k = 0.75; (D) k = 1.

in Fig. 8A where we can see that the cdf by using the classical UTSP and U-GBP distributions is always below the cdf of the beta distribution for any kind of mixture. However, by letting n vary the cdf of the U-TSP and U-GBP distributions can give a crossrepresentation of the U-BETA cdf (see Fig. 8B). To highlight this situation, the 95% percentile of the time needed to complete the project is represented in Fig. 9 for different values of k. In the case of the U-BETA distribution the percentile is constant, while in the case of the U-TSP and U-GBP distributions, the percentile depends on the value of n. Note that the parameter n is related to the peakedness. As it is shown, by using small values of n, the distributions U-TSP and U-GBP estimate (with a 95% probability) a time to complete the project that is longer than the UBETA distribution. However, by using values of n higher than approximately five, the U-TSP and U-GBP distributions estimate a time to complete the project that is shorter than the U-BETA distribution. The differences for k = 0.5 and k = 0.75 are 4 and 7 days, respectively, or similarly, a variation of 7% and 12% respectively. This is to support the importance of the parameter n and the joint elicitation of n and k by using percentiles.

5. Conclusions The mixtures of the Uniform distribution with the Two Sided Power distribution (U-TSP) and the Uniform distribution with the Generalized Biparabolic distribution (U-GBP) have been introduced in this paper as an alternative to the Uniform-Beta distribution presented by Hahn (2008). First, we compare the mixture presented by Hahn (2008) with the mixture between the uniform and classical TSP (n = 0.302344) distribution and the uniform and classical GBP (n = 2.74669) distribution. Focussing on the kurtosis coefficient, Figs. 3B and 4B show that the kurtosis coefficients of the classical U-TSP and classical U-GBP distributions are higher than the one in the classical U-BETA

distribution for 0.25 < h < 0.75 for all values of k and for k < 0:25 for all values of h. Therefore, we can conclude that the U-BETA distribution only presents a higher kurtosis in the asymmetric cases. Taking into account that the most likely value (m) given by the expert used for centering with respect to the values a and b, we can conclude that in the most of the cases the classical U-TSP and the classical U-GBP distributions will have higher kurtosis than the classical UBETA, and, as a consequence, these distributions will be more suitable for modeling heavy tails phenomena. On the other hand, the cdf of the U-TSP and U-GBP distributions present more possibilites by allowing n to vary, as is shown in Fig. 8B where the cdf of the U-TSP and U-GBP distributions cross the cdf of the U-BETA distribution. The clear consequence of this is that, with an adequate elicitacion, or, what amounts to the same, a good expert, the elicitation procedure introduced here allows us to increase the accuracy of the estimation. A new elicitation procedure is proposed as an alternative to the one proposed by Hahn (2008) which does not follow the usual methodology applied in PERT and only takes into account the classical beta (k = 4 or k = 6) and only offers the elicitation for a, m and b parameters. The U-TSP and the U-GBP distributions have a closed expression for the cumulative distribution function so they can be directly elicitated from quantiles, and to allow different values for the parameter n that can be easily elicitated. It was shown that this parameter n has a great importance in the estimation of the time needed to complete the project which is essential in PERT methodology. In addition, by increasing n, the value of kurtosis increases and the variance decreases. Therefore we can say that we obtain more accuracy and a greater likelihood of more extreme tail-area events. There are at least two areas in which this research can be extended: first, the development of new distributions with more parameters that can be applied in PERT methodology and, at the same time, complete the relative scarcity of bounded distribution in literature; second, to find more applications of these

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