delay.stochastic.pert {ProjectManagement}

Problems of distribution of delay in stochastic projects

Description

This function calculates the delay of a stochastic project, once it has been carried out. In addition, it also calculates the distribution of the delay on the different activities with the Stochastic Shapley rule.

Usage

delay.stochastic.pert(
  prec1and2 = matrix(0),
  prec3and4 = matrix(0),
  distribution,
  values,
  observed.duration,
  percentile = NULL,
  delta = NULL,
  cost.function = NULL,
  compilations = 1000
)

Arguments

prec1and2

A matrix indicating the order of precedence type 1 and 2 between the activities (Default=matrix(0)). If value (i,j)=1 then activity i precedes type 1 to j, and if (i,j)=2 then activity i precedes type 2 to j. Cycles cannot exist in a project, i.e. if an activity i precedes j then j cannot precede i.

prec3and4

A matrix indicating the order of precedence type 3 and 4 between the activities (Default=matrix(0)). If value (i,j)=3 then activity i precedes type 3 to j, and if (i,j)=4 then activity i precedes type 4 to j. Cycles cannot exist in a project, i.e. if an activity i precedes j then j cannot precede i.

distribution

Type of distribution that each activities' duration has. It can be NORMAL, TRIANGLE, EXPONENTIAL, UNIFORM, T-STUDENT, FDISTRIBUTION, CHI-SQUARED, GAMMA, WEIBULL, BINOMIAL, POISSON, GEOMETRIC, HYPERGEOMETRIC and EMPIRICAL.

values

Matrix with the parameters corresponding to the distribution associated with the duration for each activity. Considering i as an activity we have the following cases. If the distribution is TRIANGLE, then (i, 1) it is the minimum value, (i, 2) the maximum value and (i, 3) the mode. If the distribution is NORMAL, (i, 1) is the mean and (i, 2) the variance. If the distribution is EXPONENTIAL, then (i, 1) is the \lambda parameter. If the distribution is UNIFORM, (i, 1) it is the minimum value and (i, 2) the maximum value. If the distribution is T-STUDENT, (i, 1) degrees of freedom and (i, 2) non-centrality parameter delta. In FDISTRIBUTION, (i, 1) and (i, 2) degrees of freedom and (i, 3) non-centrality parameter. In CHI-SQUARED, (i, 1) degrees of freedom and (i, 2) non-centrality parameter (non-negative). In GAMMA, (i, 1) and (i, 3) shape and scale parameters and (i, 2) an alternative way to specify the scale. In WEIBULL, (i, 1) and (i, 2) shape and scale parameters. In BINOMIAL, (i, 1) number of trials (zero or more) and (i, 2) probability of success on each trial. In POISSON, (i, 1) non-negative mean. In GEOMETRIC, (i, 1) probability of success in each trial, between 0 and 1. In HYPERGEOMETRIC, (i, 1) number of white balls in the urn, (i, 2) number of black balls in the urn and (i, 3) numer of balls drawn from the urn. Finally, if the distribution is EMPIRICAL, then (i,j), for all j\in \{1,...,M\} such that M>0, is the sample.

observed.duration

Vector with the observed duration for each activity.

percentile

Percentile used to calculate the maximum time allowed for the duration of the project (Default=NULL). Only percentile or delta is necessary. This value is only used if the function uses the default cost function.

delta

Maximum time allowed for the duration of the project (Default=NULL). Only delta or pencetile is necessary. 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.

compilations

Number of compilations that the function will use for average calculations (Default=1000).

Details

Given a problem of sharing delays in a stochastic project (N,\prec,\{X_i\}_{i\in N},\{x_i\}_{i\in N}), such that \{X_i\}_{i\in N} is the random variable of activities' durations and \{x_i\}_{i\in N} the observed value. It is defined as E(D(N,\prec,\{X_i\}_{i\in N})) the expected project time, where E is the mathematical expectation, and D(N,\prec,\{x_i\}_{i\in N}) the observed project time, then d=D(N,\prec,\{X_i\}_{i\in N})-\delta, with \delta>0, normally \delta>E(D(N,\prec,\{X_i\}_{i\in N})), is the delay. The proportional and truncated proportional rule, see delay.pert function, can be adapted to this context by using the mean of the random variables.

The Stochastic Shapley, Gonçalves-Dosantos et al. (2020), rule is based on the Shapley value for the TU game (N,v) where v(S)=E(C(D(N,\prec,(\{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(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).

The Stochastic Shapley rule 2 is based on the sum of the Shapley values for the TU games (N,v) and (N,w) where v(S)=E(C(D(N,\prec,(\{X_i\}_{i\in N\backslash S},\{x_i\}_{i\in S}))))-E(C(D(N,\prec,(\{X_i\}_{i\in N})))) and w(S)=E(C(D(N,\prec,(\{0_i\}_{i\in N\backslash S},\{X_i\}_{i\in S})))), for all S\subseteq N, 0_N denotes the vector in R^N whose components are equal to zero and where C is the costs function (by default C(y)=D(N,\prec,y)-\delta).

Value

A delay value and solution vector.

References

castro

Castro, J., Gómez, D., & Tejada, J. (2009). Polynomial calculation of the Shapley value based on sampling. Computers & Operations Research, 36(5), 1726-1730.

gon

Gonçalves-Dosantos, J.C., García-Jurado, I., Costa, J. (2020) Sharing delay costs in Stochastic scheduling problems with delays. 4OR, 18(4), 457-476

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)
distribution<-c("TRIANGLE","TRIANGLE","TRIANGLE","TRIANGLE","EXPONENTIAL")
values<-matrix(c(1,3,2,1/2,3/2,1,1/4,9/4,1/2,3,5,4,1/2,0,0),nrow=5,byrow=TRUE)
observed.duration<-c(2.5,1.25,2,4.5,3)
percentile<-NULL
delta<-6.5
delay.stochastic.pert(prec1and2=prec1and2,distribution=distribution,values=values,
observed.duration=observed.duration,percentile=percentile,delta=delta,
cost.function=NULL,compilations=1000)