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```
#' Computes the minimum and maximum money pump index
#'
#' The nonparametric GARP test is "sharp" in nature. This means that the test will only tell us whether the observed data
#' set is consistent with GARP. However, when the data set fails, it is often useful to know how close the observed behavior is to
#' satisfying the rationality restrictions (see Varian (1990) for an extensive motivation). Over the years, several measures (called
#' goodness-of-fit indices) have been introduced to evaluate the degree to which the observed data set is consistent with the
#' rationality axiom. Echenique et al. (2011) proposed the money pump index (MPI) as a measure of the severity of a GARP violation.
#' The MPI is defined as the amount of money that an arbitrageur can pump from the consumer. The higher is
#' the MPI value, the more severe is the violation of rationality. While the MPI measure is conceptually appealing, it may
#' be computationally challenging to determine this index for data sets with a large number of observations. In particular,
#' Smeulders et al. (2013) showed that computing the mean and median MPI is an NP-hard problem. As easy-to-apply
#' alternatives, they proposed the minimum and maximum MPI which can be computed efficiently (in polynomial time).
#' This function implements the algorithm provided by these authors to measure the minimum and maximum MPI values for the
#' given data set.
#'
#' @param p A \eqn{T X N} matrix of observed prices where each row corresponds to an observation and each column
#' corresponds to a consumption category. \eqn{T} is the number of observations and \eqn{N} is the number of consumption
#' categories.
#'
#' @param q A \eqn{T X N} matrix of observed quantities where each row corresponds to an observation and each column
#' corresponds to a consumption category.\eqn{T} is the number of observations and \eqn{N} is the number of consumption
#' categories.
#'
#' @return The function returns two elements. The first element (\code{minimum_MPI}) is the minimum MPI and the second
#' element (\code{maximum_MPI}) is the maximum MPI.
#'
#' @section References:
#' \itemize{
#' \item Echenique, Federico, Sangmok Lee, and Matthew Shum. "The money pump as a measure of revealed preference
#' violations." Journal of Political Economy 119, no. 6 (2011): 1201-1223.
#' \item Smeulders, Bart, Laurens Cherchye, Frits CR Spieksma, and Bram De Rock. "The money pump as a measure of
#' revealed preference violations: A comment." Journal of Political Economy 121, no. 6 (2013): 1248-1258.
#' \item Varian, Hal R. "Goodness-of-fit in optimizing models." Journal of Econometrics 46, no. 1-2 (1990): 125-140.
#' }
#'
#' @examples
#' # define a price matrix
#' p = matrix(c(4,4,4,1,9,3,2,8,3,1,
#' 8,4,3,1,9,3,2,8,8,4,
#' 1,4,1,8,9,3,1,8,3,2),
#' nrow = 10, ncol = 3, byrow = TRUE)
#'
#' # define a quantity matrix
#' q = matrix(c( 1.81,0.19,10.51,17.28,2.26,4.13,12.33,2.05,2.99,6.06,
#' 5.19,0.62,11.34,10.33,0.63,4.33,8.08,2.61,4.36,1.34,
#' 9.76,1.37,36.35, 1.02,3.21,4.97,6.20,0.32,8.53,10.92),
#' nrow = 10, ncol = 3, byrow = TRUE)
#'
#'
#' # compute the minimum and maximum MPI
#' mpi(p,q)
#'
#' @seealso \code{\link{ccei}} for the critical cost efficiency index.
#'
#' @export
#'
mpi <- function(p,q)
{
if (nrow(p)!=nrow(q)) stop("Number of observations must be the same for both price and quanity matrices")
if (ncol(p)!=ncol(q)) stop("Number of consumption categories must be the same for both price and quanity matrices")
if((dim(p)[1] != dim(q)[1]) || (dim(p)[2] != dim(q)[2])){
print("Dimensions of price and quantity matrix do not match")
}
else{
p_new = c()
q_new = c()
for(k in 1:dim(p)[1]){
if(sum(q[k,]) !=0){
p_new = rbind(p_new,p[k,])
q_new = rbind(q_new,q[k,])
}
}
d <- graph_list(p_new,q_new,-1) ## d stores the graph corresponding to the consumption data
if(cycle_detection_topo(d)== 0){return(c(0,0))} ## if graph is acyclic, then there is no GARP violation
minimum_MPI <- minimum_cost_time(d,-100000000,100000000)
d <- graph_list(p_new,q_new,1)
maximum_MPI <- -1* minimum_cost_time(d,-100000000,100000000)
return(round(c(minimum_MPI,maximum_MPI),10))
}
}
```

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