[R] lpSolve: how to allow variables to become negative

[Alicia Perez-alonso]

Dear all,

I am interested in solving a MIP problem with binary outcomes and 
continuous variables, which ARE NOT RESTRICTED TO BE NEGATIVE. In 
particular,

Max {z1,z2,z3,b1}  z1 +  z2 + z3 
(s.t.)
#   7 z1 + 0 z2 + 0 z3 + b1 <= 5
#   0 z1 + 8 z2 + 0 z3 - b1 <= 5
#   0 z1 + 0 z2 + 6 z3 + b1 <= 7
#   z1, z2, z3   BINARY {0,1}
#  -5<= b1 <=5 (i.e.  b1 <= 5; -b1 <= 5 )

Using the lpSolve package of R, I wrote:

library (lpSolve)
f.obj <- c(1, 1, 1, 0)
f.con <- matrix (c(7, 0, 0, -1, 0, 8, 0, 1, 0, 0, 6, -1, 0, 0, 0, 1 ), 
nrow=4, byrow=TRUE)
f.dir <- c("<=", "<=", "<=", "<=")
f.rhs <- c(5, 5, 7, 5)

lp ("max", f.obj, f.con, f.dir, f.rhs, binary.vec=1:3)
lp ("max", f.obj, f.con, f.dir, f.rhs, binary.vec=1:3)$solution

The problem is that BY DEFAULT,  "b1" is constrained to be ">=0",  thus, 
the constraint "-b1 <= 5" does not apply.

Given that there is not any global variable in the lp() command such 
that I can change the default value of 0 (to -5 or -inf), I do not get 
the correct solution to the problem, which is indeed b1= -2.

Thank you very much in advance for your help,
Alicia

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_________________________________________________________
Alicia Pérez-Alonso
Departament of Economics
European University Institute
via della Piazzuola 43
50133 Firenze (FI)
ITALY

Room: 40 (Second Floor)
Phone: (+39) 0554685955
Fax: (+39) 0554685902
E-mail: alicia.perez-alonso at eui.eu
URL: http://www.eui.eu/Personal/Fellows/Perez-Alonso/