| delay.pert {ProjectManagement} | R Documentation |
Problems of distribution of delay in deterministic projects
Description
This function calculates the delay of a project once it has been completed. In addition, it also calculates the distribution of the delay between the different activities with the proportional, truncated proportional and Shapley rule.
Usage
delay.pert(
duration,
prec1and2 = matrix(0),
prec3and4 = matrix(0),
observed.duration,
delta = NULL,
cost.function = NULL
)
Arguments
duration |
Vector with the expected duration for each activity. |
prec1and2 |
A matrix indicating the order of precedence type 1 and 2 between the activities (Default=matrix(0)). If value |
prec3and4 |
A matrix indicating the order of precedence type 3 and 4 between the activities (Default=matrix(0)). If value |
observed.duration |
Vector with the observed duration for each activity. |
delta |
Value to indicate the maximun time that the project can take without delay. If this is not added, the function will use as delta the expected project time. This value is only used if the function uses the default cost function. |
cost.function |
Delay costs function. If this value is not added, a default cost function will be used. |
Details
Given a problem of sharing delays in a project (N,\prec,\{\bar{X}_i\}_{i\in N},\{x_i\}_{i\in N}), such that \{\bar{X}_i\}_{i\in N} is the expected value of activities' duration and \{x_i\}_{i\in N} the observed value. If D(N,\prec,\{\bar{X}_i\}_{i\in N}) is the expected project time and D(N,\prec,\{x_i\}_{i\in N}) is the observed project time, it has to d=D(N,\prec,\{\bar{X}_i\}_{i\in N})-\delta is the delay, where \delta can be any arbitrary value greater than zero. The following rules distribute the delay costs among the different activities.
The proportional rule, from Brânzei et al. (2002), distributes the delay, d, proportionally. So that each activity receives a payment of:
\phi_i=\frac{\displaystyle x_{i}-\bar{X}_{i}}{\displaystyle \sum_{j\in N}\max\{x_{j}-\bar{X}_{j},0\}}\cdot C(D(N,\prec,\{\bar{X}_i\}_{i\in N})).
The truncated proportional rule, from Brânzei et al. (2002), distributes the delay, d, proportionally, where the individual delay of each player is reduced to d if if is larger. So that each activity receives a payment of:
\bar{\phi}_i=\frac{\displaystyle \min\{x_{i}-\bar{X}_{i},C(D(N,\prec,\{\bar{X}_i\}_{i\in N}))\}}{\displaystyle \sum_{j\in N} \max\{\min\{x_{j}-\bar{X}_{j},C(D(N,\prec,\{\bar{X}_i\}_{i\in N}))\},0\}}\cdot C(D(N,\prec,\{\bar{X}_i\}_{i\in N})).
These values are only well defined when the sum of the individual delays is different from zero.
Shapley rule distributes the delay, d, based on the Shapley value for TU games, see Bergantiños et al. (2018). Given a project problem with delays (N,\prec,\{\bar{X}_i\}_{i\in N},\{x_i\}_{i\in N}), its associated TU game, (N,v), is v(S)=C(D(N,\prec,(\{\bar{X}_i\}_{i\in N\backslash S},\{x_i\}_{i\in S}))) for all S\subseteq N, where C is the costs function (by default C(D(N,\prec,y))=D(N,\prec,y)-\delta. If the number of activities is greater than ten, the Shapley value, of the game (N,v), is estimated using a unique sampling process for all players, see Castro et al. (2009).
Value
The delay value and a solution matrix.
References
- berg
Bergantiños, G., Valencia-Toledo, A., & Vidal-Puga, J. (2018). Hart and Mas-Colell consistency in PERT problems. Discrete Applied Mathematics, 243, 11-20.
- bran
Brânzei, R., Ferrari, G., Fragnelli, V., & Tijs, S. (2002). Two approaches to the problem of sharing delay costs in joint projects. Annals of Operations Research, 109(1-4), 359-374.
- castro
Castro, J., Gómez, D., & Tejada, J. (2009). Polynomial calculation of the Shapley value based on sampling. Computers & Operations Research, 36(5), 1726-1730.
Examples
prec1and2<-matrix(c(0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0),nrow=5,ncol=5,byrow=TRUE)
duration<-c(2,1,1,4,2)
observed.duration<-c(2.5,1.25,2,4.5,3)
delta<-6
delay.pert(duration,prec1and2=prec1and2,observed.duration=observed.duration,
delta=delta,cost.function=NULL)