On The Optimal Management Of Project Risk

European Journal of Operational Research 107 (1998) 451±469

On the optimal management of project risk L. Valadares Tavares *, J.A. Antunes Ferreira 1, J. Silva Coelho

1

CESUR-IST, Av. Rovisco Pais, 1000 Lisboa, Portugal Received 1 July 1996; received in revised form 1 March 1997

Abstract The uncertainty of project networks has been mainly considered as the randomness of duration of the activities. However, another major problem for project managers is the uncertainty due to the randomness of the amount of resources required by each activity which can be expressed by the randomness of its cost. Such randomness can seriously a€ect the discounted cost of the project and it may be strongly correlated with the duration of the activity.In this paper, a model considering the randomness of both the cost and the duration of each activity is introduced and the problem of project scheduling is studied in terms of the project's discounted cost and of the risk of not meeting its completion time. The adoption of the earliest (latest) starting time for each activity decreases (increases) the risk of delays but increases (decreases) the discounted cost of the project. Therefore, an optimal compromise has to be achieved. This problem of optimization is studied in terms of the probability of the duration and of the discounted cost of the project falling outside the acceptable domain (Risk function) using the concept of ¯oat factor as major decision variable. This last concept is proposed to help the manager to synthetize the large number of the decision variables representing each schedule for the studied project. Numerical results are also presented for a speci®c project network. Ó 1998 Elsevier Science B.V. All rights reserved. Keywords: Project management; Scheduling; Risk analysis; Finance

1. The concept of project and of project risk The concept of project has evolved since the early papers about networking and project scheduling (see e.g., [1]), considered nowadays as an interconnected set of activities aiming to achieve speci®c goals which can be related to a wide range of objectives, such as the development of a new * Corresponding author. Fax: +351 8409884; e-mail: [email protected]. 1 Fax: +351 8409884; e-mail: [email protected].

0377-2217/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 7 - 2 2 1 7 ( 9 7 ) 0 0 3 4 4 - 5

product, the implementation of an organizational change or the construction of a new building [2]. This implies that each project is supposed to attain speci®ed levels of indicators which should express the goals of the institution responsible for the project. These indicators focus relevant perspectives such as ± duration ± cost ± bene®t ± consumed resources ± quality of the results,

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and quite often, these perspectives can be aggregated into three major indicators: Z ± time (duration, milestones, etc.); X ± cost (resources, expenses, etc.); Y ± bene®t (outputs, quality, receivals, etc.). Thus, the performance of the project will be de®ned in terms of these three indicators and a ``target domain'' will be de®ned by f X 6 LX ; Y P LY ; Z 6 LZ g: The bene®ts are due to the use of the system (or of the changes) produced by the project and they tend to be more dependent on external factors (market, demand, competition, etc.) rather than on the internal development of the project, at least if the achieved level of quality is assumed to be constant. In this case, the study of the project's performance can be carried out in terms of the plane XOZ (Fig. 1) and the target domain is de®ned by X. Whenever the performance falls outside X, the project fails to meet its objectives and the probability of such an event can be considered as a measure of the project risk. This risk depends on the uncertainty of the project's components, namely, the randomness of the duration and of the cost of each activity, as well as the calendar adopted to schedule the project's activities. Research on uncertainty has focused on the issue of duration but has not given enough attention to the randomness of the costs [3]. Also, the duration and cost of each activity are usually considered independent variables but unfortunately this is not the case of most real projects. Actually, a longer duration of an activity is usually due to the need of carrying out additional works or of overcoming unforeseen problems and therefore,

Fig. 1. Reduced target domain.

additional resources are also required by such activity. The research presented in this paper is oriented to study the project's risk as a function of the uncertainty of the duration and the cost of each activity and in terms of the adopted schedule which is considered the major decision of this problem. The adoption of the earliest (latest) starting time for each activity reduces (increases) the risk of an overall delay but increases (decreases) the project's discounted cost and therefore an optimal compromise has to be achieved [4]. Unfortunately, the decision on the schedule implies the selection of a large number of decision variables which is equal to the number of activities. A concept is proposed in the next section ± ¯oat factor ± to help the project manager through the synthesis of that large set of variables into a single decision variable. Then, the optimal value for this factor is studied in terms of a stochastic model describing the project network. 2. On the synthesis of the project's schedule: A theorem about the propagation of ¯oats 2.1. The problem One of the most critical problems in Project Management concerns the use of the ¯oats of the activities to schedule a project. Actually, scheduling a project implies setting up the starting time, tis of each activity, i ˆ 1; . . . ; N and by de®nition, each time tis should satisfy (a) tis …E† 6 tis 6 tis …L†; where tis …E† and tis …L† are the earliest and the latest starting times of i, and (b) tis ‡ di 6 minj2J …i† tjs ; where di is the duration of i and tis ; tjs are the adopted schedule times for i and j. The set J(i) just includes all the activities requiring the immediate precedence of i as it is shown in Fig. 2 (using the usual convention Activity-on-Arc which is adopted in this paper).

Fig. 2. The precedence relationship.

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Fig. 3. Structure of the decision-aid model.

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The di€erence between tis … L† and tis … E† is called the total ¯oat of i,

variables (N starting times) and to the size of the set of constraints to be satis®ed. In Section 2.2, a synthetic description of such decisions is proposed in order that their evaluation and selection will be performed by easier procedures.

Di ˆ tis … L† ÿ tis … E†: The adoption of tis near tis … L† increases the risk of overall delays but it decreases the discount cost of the project and the other extreme policy of making tis near tis … E† has opposite e€ects. Usually, the most convenient policy to manage the ¯oat Di avoids these two extremes but its study is rather complex due to the too large number of decisions

2.2. A theorem on the propagation of ¯oats The proposed synthesis is described by the factor a (¯oat factor) with

Fig. 4. The network under study (activities i ˆ 1; . . . ; 16).

Table 1 Case A Activity 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Duration

Cost

Mode

l

Lower limit

r

c

l

r

58.26 59.13 129.57 81.74 41.74 69.57 8.70 20.87 43.48 41.74 6.09 32.17 77.39 67.83 56.52 39.13

67.00 75.00 149.00 94.00 48.00 86.00 10.00 24.00 50.00 54.00 7.00 37.00 89.00 85.00 65.00 45.00

50.66 51.42 112.67 71.08 36.29 60.49 7.56 18.15 37.81 36.29 5.29 27.98 67.30 58.98 49.15 34.03

3.65 4.98 5.44 4.32 3.09 5.04 1.41 2.19 3.15 4.40 1.18 2.71 4.21 5.17 3.60 2.99

2.99 4.32 2.99 2.99 2.99 3.97 2.98 2.99 2.99 4.58 2.97 3.00 2.99 4.16 2.99 2.99

151.00 34.00 219.00 126.00 220.00 97.00 34.00 103.00 170.00 51.00 20.00 207.00 154.00 194.00 292.00 69.00

7.07 4.47 5.92 5.48 9.49 8.37 3.87 6.32 7.94 5.57 3.16 8.94 10.00 10.95 10.00 8.94

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tis …a† ˆ tis … E† ‡ a:Di ;

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activity j 2 J …i†. Therefore the following theorem should be proved.

where 0 6 a 6 1. The latest (earliest) schedule is obtained with a ˆ 1…a ˆ 0†. In order to use tis …a† with the same a for the whole network it is necessary to prove that the ®nishing time of i assuming tis …a† is not incompatible with adopting tjs …a† for any

Theorem of the ¯oat factor. The ®nishing time of i assuming that i starts at tis …a† is not greater than the starting time of any j using the same a, tjs …a†,

Table 2 Case B Activity 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Duration

Cost

Mode

l

Lower limit

r

c

l

r

51.07 51.71 113.58 71.66 36.58 60.86 7.62 18.30 38.12 36.49 5.33 28.21 67.85 59.33 49.55 34.31

67.00 75.00 149.00 94.00 48.00 86.00 10.00 24.00 50.00 54.00 7.00 37.00 89.00 85.00 65.00 45.00

50.66 51.42 112.67 71.08 36.29 60.49 7.56 18.15 37.81 36.29 5.29 27.98 67.30 58.98 49.15 34.03

7.31 9.97 10.90 8.64 6.20 10.07 2.83 4.35 6.29 8.77 2.39 5.41 8.40 10.31 7.19 5.97

44.64 87.54 44.71 44.29 45.19 74.82 45.49 43.75 44.08 94.98 47.67 43.97 44.22 80.38 44.40 43.93

151.00 34.00 219.00 126.00 220.00 97.00 34.00 103.00 170.00 51.00 20.00 207.00 154.00 194.00 292.00 69.00

14.14 8.94 11.83 10.95 18.97 16.73 7.75 12.65 15.87 11.14 6.32 17.89 20.00 21.91 20.00 17.89

Table 3 Schedule for the studied network Activity

Precedences

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

) ) ) ) 1 1; 2 2; 3 3; 4 5 5; 6 6; 7 7; 8 9 9; 10 10; 11 11; 12

Time ES

EF

LS

LF

Start

Finish

0.00 0.00 0.00 0.00 67.00 75.00 149.00 149.00 115.00 161.00 161.00 173.00 165.00 215.00 215.00 210.00

67.00 75.00 149.00 94.00 115.00 161.00 159.00 173.00 165.00 215.00 168.00 210.00 254.00 300.00 280.00 255.00

8.00 0.00 45.00 100.00 113.00 75.00 208.00 194.00 161.00 161.00 228.00 218.00 211.00 215.00 235.00 255.00

75.00 75.00 194.00 194.00 161.00 161.00 218.00 218.00 211.00 215.00 235.00 255.00 300.00 300.00 300.00 300.00

0.00 0.00 0.00 0.00 67.00 75.00 149.00 149.00 115.00 161.00 161.00 173.00 165.00 215.00 215.00 210.00

67.00 75.00 149.00 94.00 115.00 161.00 159.00 173.00 165.00 215.00 168.00 210.00 254.00 300.00 280.00 255.00

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with j 2 J(i). This means that tis …a† ‡ di 6 tjs …a†; with j 2 J …i† and 0 6 a 6 1: Proof. The earliest start time of any activity j; with j 2 J …i†; tjs … E† is given by tjs … E† ˆ tis … E† ‡ di ‡ di with di P 0 in order to account for other paths converging at the starting node of j. s The hlatest start time i of i; ti … L† is given by s s ti … L† ˆ minj 2 J …i† tj … L† ÿ di and the ®nishing   time of i adopting tis …a† is given by tis …a† ‡ di : Then,

h i tjs …a† ˆ tjs …E† ‡ a tjs … L† ÿ tjs …E† ˆ …1 ÿ a† tjs …E† ‡ a tjs …L†   ˆ …1 ÿ a† tis …E† ‡ di ‡ di ‡ a tjs …L† ˆ …1 ÿ a†tis … E† ‡ …1 ÿ a† di ‡ a tjs …L† ‡ …1 ÿ a†di and   tis …a† ‡ di ˆ tis …E† ‡ a tis …L† ÿ tis …E† ‡ di

Fig. 5. Calendar for a ˆ 0. Notation: h earliest schedule; n latest schedule; M critical activity.

Fig. 6. Calendar for a ˆ 1. Notation: h earliest schedule; n latest schedule; M critical activity.

L.V. Tavares et al. / European Journal of Operational Research 107 (1998) 451±469

ˆ …1 ÿ a†tis …E† ‡ atis …L† ‡ di   s s ˆ …1 ÿ a†ti … E† ‡ a min tj … L† ÿ di ‡ di j2J …i†

ˆ …1 ÿ

a†tis … E†

‡ …1 ÿ a†di ‡ a min tjs … L†: j2J …i†

Finally, tis …a† ‡ di 6 tjs …a† because a minj2J…i† tjs … L† 6 atjs … L† ‡ …1 ÿ a†di :

457

The deduced result allows the project manager to select the same ¯oat factor a for the whole set of activities. 3. A stochastic model of the project network The adopted model assumes that the duration of each activity is lognormally distributed because

Fig. 7. Distribution of s with a ˆ 0 for (a) Case A and (b) Case B.

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the lognormal law has properties quite convenient to model the distribution of the duration of each activity: (a) There is a lower bound which corresponds to the minimal feasible duration.This limit is due to contractual or technical reasons. (b) The lognormal distribution is an asymmetric distribution with the mode on the left side of the expected value, which is a very common feature

of the duration of the activities. The skewness coecient is positive and denoted by c. (c) The upper quantiles are unbounded. Actually, the occurrence of uncertain and inconvenient factors can always delay even further, the duration of the activity. Other authors have discussed and selected the lognormal distribution for project networks (see, e.g. [5]).

Fig. 8. Distribution of s with a ˆ 0.5 for (a) Case A and (b) Case B.

L.V. Tavares et al. / European Journal of Operational Research 107 (1998) 451±469

The cost of each activity is assumed to be Gaussian and the hypothesis of signi®cant interdependence between the cost and the duration is adopted as this is the most common case in real projects, where a longer duration means additional work to be carried out or problems to be overcome, always requiring additional resources. Thus, the correlation between the cost and the duration, q, is given and a linear regression is used to generate the cost in terms of the duration.

459

The project manager has to make a decision about the starting time of each activity and such time can be expressed in terms of a multiplied by the ¯oat of i, Di , determined by assuming that the durations of the activities are equal to their expected values. These scheduled starting times are denoted by tis …a† and given by tis …a† ˆ tis … E† ‡ aDi : Therefore, a simulation model can easily generate a random set of realizations …k ˆ 1; . . . ; K† of the studied project network, de®ning each

Fig. 9. Distribution of s with a ˆ 1 for (a) Case A and (b) Case B.

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realization by a duration and a cost for each activity ‰di …k †; Ci …k †Š; with i ˆ 1; . . . ; N : After selecting the set of values ftis …a†; with i ˆ 1; . . . ; N g, the project manager can start the  project and the real set of durations and costs di …k †; Ci …k †; with i ˆ 1; . . . ; N g will be considered as a realization of the adopted model. The occurred durations fdi …k†; i ˆ 1; . . . ; N and k ˆ 1; . . . ; Kg imply that for each i and k, one may have tjs …a† for j 2 J …i† not feasible for one or more

activities. Actually, the earliest starting time of any activity j for such a set of occurred durations s in realization  s k can be determined by tj …E†k ˆ maxi2I…j† ti …a†k ‡ di …k† ; where I(j) represents the set of all activities directly precedent to j and tis …a†k is the real starting time of i which is given by tis …a†k ˆ tis …E†k

if

tis …E†k > tis …a†

and

Fig. 10. Distribution of v with a ˆ 0 for (a) Case A and (b) Case B.

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tis …a†k ˆ tis …a†

if

tis … E†k 6 tis …a†:

Thus, a progressive iterative procedure starting with the activities, i, having  an empty I(i), will determine the whole set of tis …a†k ; i ˆ 1; . . . ; N and k ˆ 1; . . . ; Kg: Therefore, for each realization k, the total duration and the discounted cost can be computed. The support to the process of decision making concerning the selection of a can then be devel-

461

oped in terms of the risk estimated from the K generated realizations of the network, K being a suciently large number. This risk function, R, is the probability of falling outside the target domain de®ned by (Lx , Lz ). This probability is estimated by the relative number of realizations with results outside the adopted target domain. For each target domain, the optimal a is its value minimizing R. The study of the total duration is presented in terms of sk ˆ Tk =T0 where Tk is the

Fig. 11. Distribution of v with a ˆ 0.5 for (a) Case A and (b) Case B.

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total duration correspondent to the realization k and T0 is the total deterministic network duration, assuming that the duration of each activity is equal to its mean. The analysis of the discounted cost can be made for each k, in terms of the Present Cost, PC, which is given by X s Ci …k †f ti …a†k ; PCk ˆ i

where f is the discount factor, and it is assumed that the cost of each activity, Ci (k), is allocated to its starting time, tis …a†k . Then, PC will be studied in relative terms by P s Ci …k †f ti …a†k ; vk ˆ P tis …a0 † i l…Ci † f where l…Ci † is the average of Ci …k†, and a0 is the reference value adopted for a…usually; a0 ˆ 0†.

Fig. 12. Distribution of v with a ˆ 1 for (a) Case A and (b) Case B.

L.V. Tavares et al. / European Journal of Operational Research 107 (1998) 451±469

The estimated v for the realization k is denoted by vk . Therefore, the target domain will be de®ned in terms of X ˆ v and Z ˆ s. Alternatively, the study of the cost can be done in terms of the Relative Present Value of the project, RPV, which is de®ned for the realization k by ," # X s Ci …k †f ti …a†k ; RPVk ˆ …k † ˆ … BfTk † i

463

where B is the bene®t received when the project is completed. The RPV obtained for the deterministic network with the duration and cost of each activity equal to their means, and with a ˆ a0 , is denoted by RPV0 , and so the estimated results obtained from the simulation can be presented for each k through wk ˆ

RPVk ; RPV0

Fig. 13. Estimated mean and quantitiles of s in terms of a for (a) Case A (b) and Case B.

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where RPVk is the RPV computed for the generated realization k. It should be noted that w is not a function of B because wk ˆ f

Tk ÿT0

P s Ci …k †f ti …a†k Pi : tis i l…Ci †f …a0 †

In this case, X would be made equal to w. This model is implemented as a decision aid to help the project manager to select the most

appropriate a in terms of s and v, using the structure presented in Fig. 3. 4. An application 4.1. Results The developed model was applied to an illustrative project network with the data presented in

Fig. 14. Estimated mean and quantitiles of v in terms of a for (a) Case A and (b) Case B.

L.V. Tavares et al. / European Journal of Operational Research 107 (1998) 451±469

Fig. 4 and in Table 1 (Case A). The parameter q is assumed to be equal to 0.7. Alternatively, a network with a higher uncertainty is also studied (Case B) multiplying the previous standard deviation of the duration and of the cost by a factor of 2 (Table 2). The major decision variable is a which varies between 0 and 1 and the risk function, R, is studied in terms of the limits of the target domain …Lx; Lz†, as was previously presented. The study of this network is initially performed assuming deterministic data equal to the means of durations and costs. The result for the start and ®nish times are presented in Table 3. The calendar is presented in Figs. 5 and 6 for a ˆ 0 and a ˆ 1, obtaining a total duration of 300 units.

465

The stochastic simulation is carried out using the presented model. A sample of 10 ´ 1000 realizations is generated and, for Cases A and B, the estimated distribution of s and v (where p‰sŠ or p‰vŠ denotes the estimated probability of occurrence of s or v within the corresponding interval) are presented in Figs. 7±12 for a ˆ 0, a ˆ 0.5 and a ˆ 1.0. The analysis of s and v in terms of a is also presented in Figs. 13 and 14 where the estimated mean (l) and the 5% and 95% quantiles (Q0:05 , Q0:95 ) are plotted. The estimated results for the average, standard deviation and quantiles of s and v in terms of a and of the adopted case are presented in Tables 4 and 5. In these tables, the standard error of such

Table 4 Estimated parameters of the distribution of s a

0.0

Variance

LV

HV

0.5 LV

HV

LV

1.0 HL

l SE(l) r SE(r) Q0:95 SE(Q0:95 ) Q0:05 SE(Q0:05 )

1.074 0.001 0.146 0.003 1.345 0.007 0.940 0.001

1.266 0.011 0.833 0.134 2.226 0.037 0.917 0.000

1.081 0.001 0.143 0.003 1.350 0.005 0.952 0.000

1.275 0.010 0.870 0.240 2.237 0.035 0.925 0.000

1.102 0.001 0.132 0.004 1.348 0.006 0.990 0.001

1.306 0.007 0.782 0.099 2.279 0.030 0.968 0.000

LV ± Low variance (Case A). HV ± High variance (Case B).

Table 5 Estimated parameters of the distribution of v a

0.0

Variance

LV

HL

LV

HV

LV

HL

l SE(l) r SE(r) Q0:95 SE(Q0:95 ) Q0:05 SE(Q0:05 )

0.969 0.001 0.028 0.001 1.018 0.001 0.928 0.001

0.957 0.002 0.148 0.023 1.101 0.006 0.819 0.001

0.811 0.000 0.026 0.000 0.855 0.001 0.765 0.019

0.797 0.001 0.128 0.012 0.913 0.005 0.718 0.001

0.686 0.001 0.025 0.001 0.728 0.001 0.648 0.000

0.675 0.001 0.110 0.012 0.780 0.004 0.603 0.001

LV ± Low variance (Case A). HV ± High variance (Case B).

0.5

1.0

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Fig. 15. Estimated risk for Region 1 in terms of a for (a) Case A and (b) Case B.

parameters (SE ( )) are estimated using 10 samples of 1000 realizations. Finally, the risk function, R, was computed (for Cases A and B) in terms of a and using di€erent bounds for Lx and Lz :

Region 1 Region 2 Region 3

v

s

Lx ˆ 0.6 Lx ˆ 0.8 Lx ˆ 1.0

Lz ˆ 1.0 Lz ˆ 1.20 Lz ˆ 1.40

This set of results is presented in Figs. 15±17, allowing the estimation of the optimal a which is given by the a minimizing R. Therefore, the recommended values for a are given in Table 6. 4.2. Comments The presented results deserve the following comments:

L.V. Tavares et al. / European Journal of Operational Research 107 (1998) 451±469

467

Fig. 16. Estimated risk for Region 2 in terms of a for (a) Case A and (b) Case B.

± The distribution of s always has a positive skewness, showing that the Gaussian assumption of PERT is far from acceptable for this type of example. However, the distribution of v is reasonably symmetrical. It should be noted that these results were obtained assuming a lognormal law for the duration of the activities and a linear Gaussian regression for the cost in terms of the duration. ± The level of variance adopted is crucial as the assumption of a higher variance for the durations of the activities increases the e€ect of a very significantly. Actually, the increase of the duration and

the decrease of the present cost is much clearer when a increases for Case B than for Case A. This e€ect is particularly strong for the skewness and the upper quantile. ± The presented results for the risk, R, show that the concept of optimal a can be estimated through the presented model. The risk gain due to adopting the optimal a grows with the uncertainty of the data as would be expected and its value also depends signi®cantly on the bounds adopted and on the level of variance (Case A or B).

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Fig. 17. Estimated risk for Region 3 in terms of a for (a) Case A and (b) Case B.

Table 6 Optimal value or interval for a Region

Case A

Case B

1 2 3

0.8 0.4±0.8 0.1±1.0

0.7 0.3±0.7 0.3±0.9

5. Conclusions The concept of project risk was proposed and de®ned in this paper as a function of the discounted

cost and duration associated with the project's network. These magnitudes are modelled in terms of the scheduled starting time of each activity and the concept of a ¯oat factor is introduced as a decision aid to help the project manager to select the most convenient schedule. The advantage of using this concept stems from the possibility of using the same factor throughout the whole project, as has been proved in this paper. A model is proposed to study the project risk in terms of the ¯oat factor (risk function).

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An application is presented showing how this model is a useful instrument to support the decisions of the project manager through the estimation of the distributions of total cost and duration and through the estimation of the project risk. The risk can easily be estimated in terms of the ¯oat factor and its optimal value can therefore be computed. References [1] J.E. Kelly, Critical path planning and scheduling mathematical bases, Operations Research 9 (1961) 246±320.

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[2] L.V. Tavares, A review on the contributions of operational research to project management, EURO 14 (1995) 67±82. [3] S.E. Elmaghraby, A guided tour through some recent developments, European Journal of Operational Research 82 (1995) 383±408. [4] L.V. Tavares, A stochastic model to control project duration and expenditure, European Journal of Operations Research 78 (1994) 262±266. [5] B. Dean, S. Mertel, Jr., L. Roepke, Research project cost distribution and budget forecasting, IEEE Transactions on Engineering Management 16 (4) (1969) 176±191.