The transportation problem is one of the classical problems teached in linear programming classes. The problem, put simply, states that a given set of customers with a specified demand must be satisfied by another set of supplier with certain capacities (“supply”). For a detailed explanation of the transportation problem you can, e.g. read this: https://econweb.ucsd.edu/~jsobel/172aw02/notes8.pdf.

The lpSolve package available in R can be used for modelling and solving the transportation problem.

I will show how to do this. I define a problem in which 3 suppliers seek to satisfy 4 customers. The suppliers have capacities 100, 300, and 400, respectively. The customers have demand 100, 100, 200, and 400, respectively. Furthermore, the cost for supplying customer i by supplier j is defined for every possible combination and stated in a cost matrix.

Using this information we can model and solve the transportation problem (deciding which demand to fulfill by which supplier) with the lpSolve package in R.

First, I prepare the modeling-part:

library(lpSolve)

# specifying cost matrix 
cost.mat <- matrix(nrow=3,ncol=4)
cost.mat[1,] <- 1:4
cost.mat[2,] <- 4:1
cost.mat[3,] <- c(1,4,3,2)

# this is a minimization problem
direction = "min"

# capacity may not be exceeded
row.signs <- rep("<=",3)
row.rhs <- c(100,300,400)

# demand must be satisfied
col.signs <- rep(">=",4)
col.rhs <- c(100,100,200,400)

Then, I solve the problem:

# solve and assign lp object
solution <- lp.transport(cost.mat = cost.mat,
                         direction = direction,
                         row.signs = row.signs,
                         row.rhs = row.rhs,
                         col.signs = col.signs,
                         col.rhs = col.rhs)

Let us review the “optimal” costs:

solution

## Success: the objective function is 1400

Let us review the optimal solution of the transportation problem (i.e., the optimal material flows to this problem):

solution$solution

##      [,1] [,2] [,3] [,4]
## [1,]    0  100    0    0
## [2,]    0    0  200  100
## [3,]  100    0    0  300

The assignment problem is another classical problem, and you can see how I solved it in R here: Cost minimal production scheduling – solving the assignment problem with lpSolve, using lp.assign.