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A stochastic model to control project duration and expenditure
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER
European Journal of Operational Research 78 (1994)262-266
A stochastic model to control project duration and expenditure L. V a l a d a r e s T a v a r e s Technical University of Lisbon, CESUR - IST, Av. Rovisco Pais, 1000 Lisbon, Portugal
Abstract
Many models of Project Management and Scheduling have recognized the non-deterministic nature of the duration of activities but, unfortunately, have adopted deterministic descriptions for resources and expenditures. In this paper, a stochastic model is proposed considering the stochastic nature of durations and expenditures as well as their statistical interdependence. Advanced analytical results are deduced to assess the financial risk of a project as well as to plan and to control its financial profiles. Keywords: Project management; Risk analysis
1. Introduction
The duration and the expenditure required to implement a project are often far from their scheduled values. The random nature of the duration of each activity has been considered and studied by many authors since the initial P E R T model was presented (Malcolm et al., 1958). However, expenditure or resources required to carry out each activity have been usually studied under a deterministic formulation which does not describe their random or stochastic nature (see, e.g., Elmaghraby, 1990, and Yang et al., 1988). This is the case of the existing software to support project management (De Wit et al., 1990). Unfortunately, the practical experience of implementing projects shows that, most often, expenditures are subject to deviations which are not less significant than those suffered by durations and, furthermore, that the importance of such deviations for project managers should receive top priority (Tavares, 1986, 1987).
2. On the randomness of durations and expenditures
The development of any activity of a project implies the fulfilment of a wide range of conditions which can affect its duration and the cost of its completion. Multiple factors causing activity delays or additional expenditures can be classified into three major groups: a) Contents. Each activity is defined in terms of a descriptive design including a list of components but the implementation of the project may lead the project manager to have to change such definition after the project being started because new problems or difficulties are discovered. This is often the case in the building industry where, for instance, changes in the design of the foundations of a building may have to be introduced due to geological information received after the construction starts. b) Resources. The non-availability of human
0377-2217/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0377-2217(94)00086-R
L.V. Tavares/ European Journal of Operational Research 78 (1994) 262-266
or material resources at the required times can increase the duration and the cost of an activity (see, e.g., Tavares, 1989). The shortage of equipment may, for instance, imply an waiting time in the development of an activity which may be also responsible for the increase of expenditure due to other permanent costs along the duration (e.g., cost of permanent manpower). c) Efficiency. The planned duration and expenditure of any activity always assumes some specific levels of productivity. However, the achieved efficiency depends strongly on internal and external conditions such as motivation, organization, professional and human qualifications, environmental conditions, social and economic turbulence, etc. This description shows that most of the factors which may be responsible for the increase of the duration of an activity can be also a cause of higher expenditures and therefore these two variables should not be described independently.
3. Proposed
model
3.1. Objectives The proposed model assumes the usual hypothesis of statistical independence between the durations of activities but, according to the previous analysis, it admits a significant correlation between the duration and the expenditure of each activity. The major result to be obtained is the deduction of the statistical properties of the time and of the expenditure required by the project and how to control these variables along its implementation. Therefore, this model can be used to support project financial planning and control through the assessment and the reduction of project uncertainty (Slowinski et al., 1989). This support can be particularly relevant for risk management. 3.2. Notation Let the following variables be defined for a network representing the studied project and
263
adopting the usual activity-on-arc formulation: - The set of the project activities, R = {Ai: i = 1 . . . . . N } , where N is the number of activities. - The probability density function of the activity A i, d i, is denoted by f ( d i) with iz i and o"i the mean and the standard deviation of di. - The probability density function of the expenditure due to the activity i, M~, is denoted by f ( M i ) , tz'i and or' being the mean and the standard deviation of M r - The sub-set of the project activities belonging to the critical path which is determined, as usually, in terms of average durations: R e . Without loss of generality, the critical activities will be numbered from 1 to M: R e = {Aj: j = 1 . . . . . M}, where the index j denotes the forward order of the critical nodes. Node j is defined by the end of the activity j and the first node of the project has index nil. These critical nodes will be denoted by C = 0 , 1 . . . . . M. - The occurrence time of the node j is denoted by Tj with mean and variance given by ETj and VT/, respectively. - The sub-set of the project activities with a finish scheduled time greater than Tj_, and equal to or smaller than Tj is R/. - The sum of expenditures due to all activities belonging to Rj is a random variable, MR j, with mean and variance given by E R r and VR/ respectively. Such expenditure, M R j, is considered allocated to node j with the occurrence time T/. - The sum of expenditures due to {R 1. . . . , R j} is denoted by SR/ with mean and variance given by ESRj and VSR/, respectively. 3.3. Formulation The classical assumption of P E R T ignores the activities outside the critical path after this path is determined. The proposed model keeps this assumption for the studied durations but it takes into account the expenditures of non-critical activities.
L.V. Tavares/ European Journal of OperationalResearch 78 (1994) 262-266
264
The durations of the activities are considered mutually independent but the expenditure and the duration of each activity are assumed correlated (Pi for Ai). Then, the expected value and the variance of T/ and of SRj can be iteratively computed by the formulae
This formula should be used iteratively from j = 1 to j = M and making p~(0)
V T j = VTj_ 1 +o1.2, ESRj = ESRj_ 1 + E R j , +
0.
The Gaussian assumption about the distribution of Rj and of T/ may be adopted in some cases as they are the sum of numerous independent variables. If so, the distribution of (SRj, Tj) is a binormal law with correlation given by Ps(J), f(SRj, Tj). Then the computation of the confidence limits can be easily performed using a standard table.
ETj = ETj_ 1 +/z j,
VSRj = VSRj_ t
=
VRj,
3.4. Resul~ for j = 1. . . . . M, and with Several important results can be deduced from the presented model: a) Distribution of the expenditure due until a certain node j. The probability of having the cumulative expenditure until node j not greater than L, FSRj(L), can be computed by
E T 0 = VT 0 = ESR 0 = VSR 0 = 0. The correlation between the cumulative expenditure, SRj, and the time spent, Tj., Ps(J), can be also deduced in terms of the correlation coefficient between MRi and dj, p(j): p,(j)
=
VSRj(L)=f+ff
CVSRj_; + VRj •~/VTj_1+ 0).2
y(SRj,
)dSRjd
E~dl~ru pf~li
.o°/
~
K, 10~
o °°
0
I 1
I 2
ooo
I J
ooo
I M -
1
I U
Crltlc:al
nod~
Fig. 1. Expenditure profiles in terms of the critical nodes.
..
L.V. Tavares/ EuropeanJournalof OperationalResearch78 (1994)262-266 The expenditure profiles in terms of the project critical nodes and for any specific exceedance risk (a) can then be computed (Fig. 1). b) Distribution of the expenditure due until a certain time. The probability of having the cumulative expenditure until time t not greater than L, FSR¢(L), can be computed by
265
TIroL t
-
M
90Y*
¢C = I Q X
FSR¢(L) = Y'. P [ S R s < L ] "P[Tj
where P[Tj-< t < Tj+I]
~q~,mur=
Fig. 3. Duration profiles in terms of the expenditure level.
=ft_ooft+_~f(dj+l) ddj+l'f(Tj) dTj and t
to L being smaller than or equal to t, FTL(t), is given by:
L f ( S R j , T/)
p[SRj<-LI = f_Jo P(rj <_t)
dSRjdTj,
M
FTL(t) = ~1 f t f(Tj) dTjP[SRj_ 1 < L < S R j ]
with
j~
--oo
with Therefore, the expenditure profiles can be determined in terms of time instead of the project critical nodes (Fig. 2). c) Distribution of the time spent to reach a certain level of expenditure, L. The probability of the time spent to reach the first node corresponding to a level of expenditure greater than or equal
prt~
.....---
O~ ,
log
--=
"nm=.
Fig. 2. Expenditure profiles in terms of time.
e[SRj_
•f ( S R j _ l ) dSRj_ 1. Then, the duration profiles for a risk a expressed in terms of the expenditure level can be obtained (Fig. 3): These three results can be used for the financial planning of the project (credits negotiation, cash-flow schedule, etc.) before the project starts or can be used to control its implementation along each stage of its development (node j given the occurrences until the node j - 1 ) . In this case, the presented distributions have to be used in their conditioned form given the information concerning the occurred part of the project (SRi_ 1 and Tj). Obviously, if the time scale of the project is sufficiently large, the summing of expenditures SRj = E~=IMR k can be done using an appropri-
L. 14. Tavares /European Journal of Operational Research 78 (1994) 262-266
266
ate factor, gg, between 0 and 1, to represent the discounting effect, J
SRy= ~ g k - M R k , k=l
but the presented results are still applicable.
4. Conclusions
A stochastic model representing the randomness of durations and of expenditures as well as their interdependence was deduced obtaining key results for risk assessment as well as for financial planning and control of a project. These results can be analytically computed without having to use Monte-Carlo techniques.
References De Wit, J., and Herroelen, W. (1990), "An evaluation of micro-computer-based software packages for project man-
agement", European Journal of Operational Research 49, 102-140, Elmaghraby, S.E. (1990), "Resource allocation via Dynamic Programming in activity networks", 2nd International Workshop on Project Management and Scheduling, Compi~gne, France. Malcom, D.G., Rosebloom, J.M., Clark, C.E., and Fazar, W. (1958), "Application of a technique for research and development program evaluation", Operational Research 7, 649-669. Slowinski, R., and Weglarz, J. (eds.) (1989), Advances in Project Scheduling, Elsevier, Amsterdam. Tavares, L.V. (1986), "Stochastic planning and control of program budgeting-the model MACAO", in J.D. Coelho and L.V. Tavares, OR Models on Microcomputers, NorthHolland, Amsterdam, 189-204. Tavares, L.V. (1987), "Optimal resource profiles for program scheduling", European Journal of Operational Research 29, 80-83. Tavares, L.V. (1989), "A multi-stage model for project scheduling under resource constraints", in: R. Slowinski and J. Weglarz (eds.), Advances in Project Scheduling, Elsevier, Amsterdam. Yang, K.K., Talbot, F.B., and Patterson, J.H. (1988), "Scheduling a project to maximize its net present value: An integer programming approach", 2nd International Workshop on Project Management and Scheduling, Compi~gne, France.
European Journal of Operational Research 78 (1994)262-266
A stochastic model to control project duration and expenditure L. V a l a d a r e s T a v a r e s Technical University of Lisbon, CESUR - IST, Av. Rovisco Pais, 1000 Lisbon, Portugal
Abstract
Many models of Project Management and Scheduling have recognized the non-deterministic nature of the duration of activities but, unfortunately, have adopted deterministic descriptions for resources and expenditures. In this paper, a stochastic model is proposed considering the stochastic nature of durations and expenditures as well as their statistical interdependence. Advanced analytical results are deduced to assess the financial risk of a project as well as to plan and to control its financial profiles. Keywords: Project management; Risk analysis
1. Introduction
The duration and the expenditure required to implement a project are often far from their scheduled values. The random nature of the duration of each activity has been considered and studied by many authors since the initial P E R T model was presented (Malcolm et al., 1958). However, expenditure or resources required to carry out each activity have been usually studied under a deterministic formulation which does not describe their random or stochastic nature (see, e.g., Elmaghraby, 1990, and Yang et al., 1988). This is the case of the existing software to support project management (De Wit et al., 1990). Unfortunately, the practical experience of implementing projects shows that, most often, expenditures are subject to deviations which are not less significant than those suffered by durations and, furthermore, that the importance of such deviations for project managers should receive top priority (Tavares, 1986, 1987).
2. On the randomness of durations and expenditures
The development of any activity of a project implies the fulfilment of a wide range of conditions which can affect its duration and the cost of its completion. Multiple factors causing activity delays or additional expenditures can be classified into three major groups: a) Contents. Each activity is defined in terms of a descriptive design including a list of components but the implementation of the project may lead the project manager to have to change such definition after the project being started because new problems or difficulties are discovered. This is often the case in the building industry where, for instance, changes in the design of the foundations of a building may have to be introduced due to geological information received after the construction starts. b) Resources. The non-availability of human
0377-2217/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0377-2217(94)00086-R
L.V. Tavares/ European Journal of Operational Research 78 (1994) 262-266
or material resources at the required times can increase the duration and the cost of an activity (see, e.g., Tavares, 1989). The shortage of equipment may, for instance, imply an waiting time in the development of an activity which may be also responsible for the increase of expenditure due to other permanent costs along the duration (e.g., cost of permanent manpower). c) Efficiency. The planned duration and expenditure of any activity always assumes some specific levels of productivity. However, the achieved efficiency depends strongly on internal and external conditions such as motivation, organization, professional and human qualifications, environmental conditions, social and economic turbulence, etc. This description shows that most of the factors which may be responsible for the increase of the duration of an activity can be also a cause of higher expenditures and therefore these two variables should not be described independently.
3. Proposed
model
3.1. Objectives The proposed model assumes the usual hypothesis of statistical independence between the durations of activities but, according to the previous analysis, it admits a significant correlation between the duration and the expenditure of each activity. The major result to be obtained is the deduction of the statistical properties of the time and of the expenditure required by the project and how to control these variables along its implementation. Therefore, this model can be used to support project financial planning and control through the assessment and the reduction of project uncertainty (Slowinski et al., 1989). This support can be particularly relevant for risk management. 3.2. Notation Let the following variables be defined for a network representing the studied project and
263
adopting the usual activity-on-arc formulation: - The set of the project activities, R = {Ai: i = 1 . . . . . N } , where N is the number of activities. - The probability density function of the activity A i, d i, is denoted by f ( d i) with iz i and o"i the mean and the standard deviation of di. - The probability density function of the expenditure due to the activity i, M~, is denoted by f ( M i ) , tz'i and or' being the mean and the standard deviation of M r - The sub-set of the project activities belonging to the critical path which is determined, as usually, in terms of average durations: R e . Without loss of generality, the critical activities will be numbered from 1 to M: R e = {Aj: j = 1 . . . . . M}, where the index j denotes the forward order of the critical nodes. Node j is defined by the end of the activity j and the first node of the project has index nil. These critical nodes will be denoted by C = 0 , 1 . . . . . M. - The occurrence time of the node j is denoted by Tj with mean and variance given by ETj and VT/, respectively. - The sub-set of the project activities with a finish scheduled time greater than Tj_, and equal to or smaller than Tj is R/. - The sum of expenditures due to all activities belonging to Rj is a random variable, MR j, with mean and variance given by E R r and VR/ respectively. Such expenditure, M R j, is considered allocated to node j with the occurrence time T/. - The sum of expenditures due to {R 1. . . . , R j} is denoted by SR/ with mean and variance given by ESRj and VSR/, respectively. 3.3. Formulation The classical assumption of P E R T ignores the activities outside the critical path after this path is determined. The proposed model keeps this assumption for the studied durations but it takes into account the expenditures of non-critical activities.
L.V. Tavares/ European Journal of OperationalResearch 78 (1994) 262-266
264
The durations of the activities are considered mutually independent but the expenditure and the duration of each activity are assumed correlated (Pi for Ai). Then, the expected value and the variance of T/ and of SRj can be iteratively computed by the formulae
This formula should be used iteratively from j = 1 to j = M and making p~(0)
V T j = VTj_ 1 +o1.2, ESRj = ESRj_ 1 + E R j , +
0.
The Gaussian assumption about the distribution of Rj and of T/ may be adopted in some cases as they are the sum of numerous independent variables. If so, the distribution of (SRj, Tj) is a binormal law with correlation given by Ps(J), f(SRj, Tj). Then the computation of the confidence limits can be easily performed using a standard table.
ETj = ETj_ 1 +/z j,
VSRj = VSRj_ t
=
VRj,
3.4. Resul~ for j = 1. . . . . M, and with Several important results can be deduced from the presented model: a) Distribution of the expenditure due until a certain node j. The probability of having the cumulative expenditure until node j not greater than L, FSRj(L), can be computed by
E T 0 = VT 0 = ESR 0 = VSR 0 = 0. The correlation between the cumulative expenditure, SRj, and the time spent, Tj., Ps(J), can be also deduced in terms of the correlation coefficient between MRi and dj, p(j): p,(j)
=
VSRj(L)=f+ff
CVSRj_; + VRj •~/VTj_1+ 0).2
y(SRj,
)dSRjd
E~dl~ru pf~li
.o°/
~
K, 10~
o °°
0
I 1
I 2
ooo
I J
ooo
I M -
1
I U
Crltlc:al
nod~
Fig. 1. Expenditure profiles in terms of the critical nodes.
..
L.V. Tavares/ EuropeanJournalof OperationalResearch78 (1994)262-266 The expenditure profiles in terms of the project critical nodes and for any specific exceedance risk (a) can then be computed (Fig. 1). b) Distribution of the expenditure due until a certain time. The probability of having the cumulative expenditure until time t not greater than L, FSR¢(L), can be computed by
265
TIroL t
-
M
90Y*
¢C = I Q X
FSR¢(L) = Y'. P [ S R s < L ] "P[Tj
where P[Tj-< t < Tj+I]
~q~,mur=
Fig. 3. Duration profiles in terms of the expenditure level.
=ft_ooft+_~f(dj+l) ddj+l'f(Tj) dTj and t
to L being smaller than or equal to t, FTL(t), is given by:
L f ( S R j , T/)
p[SRj<-LI = f_Jo P(rj <_t)
dSRjdTj,
M
FTL(t) = ~1 f t f(Tj) dTjP[SRj_ 1 < L < S R j ]
with
j~
--oo
with Therefore, the expenditure profiles can be determined in terms of time instead of the project critical nodes (Fig. 2). c) Distribution of the time spent to reach a certain level of expenditure, L. The probability of the time spent to reach the first node corresponding to a level of expenditure greater than or equal
prt~
.....---
O~ ,
log
--=
"nm=.
Fig. 2. Expenditure profiles in terms of time.
e[SRj_
•f ( S R j _ l ) dSRj_ 1. Then, the duration profiles for a risk a expressed in terms of the expenditure level can be obtained (Fig. 3): These three results can be used for the financial planning of the project (credits negotiation, cash-flow schedule, etc.) before the project starts or can be used to control its implementation along each stage of its development (node j given the occurrences until the node j - 1 ) . In this case, the presented distributions have to be used in their conditioned form given the information concerning the occurred part of the project (SRi_ 1 and Tj). Obviously, if the time scale of the project is sufficiently large, the summing of expenditures SRj = E~=IMR k can be done using an appropri-
L. 14. Tavares /European Journal of Operational Research 78 (1994) 262-266
266
ate factor, gg, between 0 and 1, to represent the discounting effect, J
SRy= ~ g k - M R k , k=l
but the presented results are still applicable.
4. Conclusions
A stochastic model representing the randomness of durations and of expenditures as well as their interdependence was deduced obtaining key results for risk assessment as well as for financial planning and control of a project. These results can be analytically computed without having to use Monte-Carlo techniques.
References De Wit, J., and Herroelen, W. (1990), "An evaluation of micro-computer-based software packages for project man-
agement", European Journal of Operational Research 49, 102-140, Elmaghraby, S.E. (1990), "Resource allocation via Dynamic Programming in activity networks", 2nd International Workshop on Project Management and Scheduling, Compi~gne, France. Malcom, D.G., Rosebloom, J.M., Clark, C.E., and Fazar, W. (1958), "Application of a technique for research and development program evaluation", Operational Research 7, 649-669. Slowinski, R., and Weglarz, J. (eds.) (1989), Advances in Project Scheduling, Elsevier, Amsterdam. Tavares, L.V. (1986), "Stochastic planning and control of program budgeting-the model MACAO", in J.D. Coelho and L.V. Tavares, OR Models on Microcomputers, NorthHolland, Amsterdam, 189-204. Tavares, L.V. (1987), "Optimal resource profiles for program scheduling", European Journal of Operational Research 29, 80-83. Tavares, L.V. (1989), "A multi-stage model for project scheduling under resource constraints", in: R. Slowinski and J. Weglarz (eds.), Advances in Project Scheduling, Elsevier, Amsterdam. Yang, K.K., Talbot, F.B., and Patterson, J.H. (1988), "Scheduling a project to maximize its net present value: An integer programming approach", 2nd International Workshop on Project Management and Scheduling, Compi~gne, France.