I have tried to use the R package LPSolve and in particular the lp.transport function to solve a optimisation problem. In my fictitious example below I have 5 office sites that I need to resource with a minimum number of employees and I have set up a cost matrix that determines the distance from each employees home to the office. I want to minimize the total distance traveled to work whilst meeting the minimum number of employees per office.
Initially this was working as I was treating all employees as equal (1). however problems have started to occur when I rate each employee by how efficient they are. For example I now want to say that officeX needs the equivalent of 2 engineers which might be made up of 4 engineers who are 50% efficient or 1 that is 200% efficient. When I do this however the solution found will split a employee across a number of offices, what I need is a additional constraint so impose that a employee can only be at 1 Office. Anyway hopefully that is enough background here is my example:
Employee <- c("Jim","John","Jonah","James","Jeremy","Jorge")
Office1 <- c(2.58321505105556, 5.13811249390279, 2.75943834864996,
6.73543614029559, 6.23080251653027, 9.00620341764497)
Office2 <- c(24.1757667923894, 19.9990724784926, 24.3538456922105,
27.9532073293925, 26.3310994833106, 14.6856664813007)
Office3 <- c(38.6957155251069, 37.9074293509861, 38.8271000719858,
40.3882569566947, 42.6658938732098, 34.2011184027657)
Office4 <- c(28.8754359274453, 30.396841941228, 28.9595182970988,
29.2042274337124, 33.3933900645023, 28.6340025144932)
Office5 <- c(49.8854888720157, 51.9164328512659, 49.948290261029,
49.4793138594302, 54.4908258333456, 50.1487397648236)
#create CostMatrix
costMat<-data.frame(Employee,Office1, Office2, Office3, Office4, Office5)
#efficiency is the worth of employees, eg if 1 they are working at 100%
#so if for example I wanted 5 Employees
#working in a office then I could choose 5 at 100% or 10 working at 50% etc...
efficiency<-c(0.8416298, 0.8207991, 0.7129663, 1.1406839, 1.3868177, 1.1989748)
#Uncomment next line to see the working version based on headcount
#efficiency<-c(1,1,1,1,1,1)
#Minimum is the minimum number of Employees we want in each office
minimum<-c(1, 1, 2, 1, 1)
#solve problem
opSol <-lp.transport(cost.mat = as.matrix(costMat[,-1]),
direction = "min",
col.signs = rep(">=",length(minimum)),
col.rhs = minimum,
row.signs = rep("==", length(efficiency)),
row.rhs = efficiency,
integers=NULL)
#view solution
opSol$solution
# My issue is one employee is being spread across multiple areas,
#what I really want is a extra constraint that says that in a row there
# can only be 1 non 0 value.
