prologeclipse-clp

How to implement This MP problem in ECLIPSE CLP or Prolog?


I want to implement this Summations as Objective and Constraints(1-6) Could anyone help me that how I can Implement them?

OBJ: Min ∑(i=1..N)∑(j=1..N) Cij * ∑(k=1..K)Xijk

constraint : ∑(k=1..K) Yik=1 (for all i in N)


Solution

  • The following answer is specific to ECLiPSe (it uses loops, array and array slice notation, which are not part of standard Prolog).

    I assume that N and K (and presumably C) are given, and your matrices are declared as

    dim(C, [N,N]),
    dim(X, [N,N,K]),
    dim(Y, [N,K]),
    

    You can then set up the constraints in a loop:

    constraint : ∑(k=1..K) Yik=1 (for all i in N)

    ( for(I,1,N), param(Y) do
        sum(Y[I,*]) $= 1
    ),
    

    Note that the notation sum(Y[I,*]) here is a shorthand for sum([Y[I,1],Y[I,2],...,Y[I,K]]) when K is the size of this array dimension.

    For your objective, because of the nested sum, an auxiliary loop/list is still necessary:

    OBJ: Min ∑(i=1..N)∑(j=1..N) Cij * ∑(k=1..K)Xijk

    ( multifor([I,J],1,N), foreach(Term,Terms), param(C,X) do
        Term = (C[I,J] * sum(X[I,J,*]))
    ),
    Objective = sum(Terms),
    ...
    

    You then have to pass this objective expression to the solver -- the details depend on which solver you use (e.g. eplex, ic).