Mixed Integer Program with Varying Interest Rates

Question

I'm currently stuck with a MIP program where the interest rate, i, is based on the number of units produced for Housing Plan A. If the number of plan A houses sold is the highest among all four types then i=1. If the number of plan A houses sold is the second highest, then i=2 and so on up to i=4. The interest rate is basically 2i%. Not really sure how to add constraints that will represent the position of plan A houses and implement the correct interest rate in the objective function. The objective function maximizes the total profit (e.g 50,000A + 40,000B + 70,000C + 80,000D). Any ideas on how to use binary variables to represent position?

Answer 1

One way of doing this is, is using a permutation matrix p(i,j). I.e.

sets 
  i = {A,B,C,D}
  j = {1,2,3,4}

binary variable p(i,j)

# assignment constraints
sum(i,p(i,j))=1
sum(j,p(i,j))=1

# quantities sold
x(j) = sum(i, p(i,j)*x(i))
x(j) >= x(j+1)

# interest rate
r(i) = sum(j, p(i,j)*r(j))
r(j) = 2*j/100

Unfortunately the expression p(i,j)*x(i) is non-linear. With some effort we can repair this as follows:

sets 
  i = {A,B,C,D}
  j = {1,2,3,4}

binary variable p(i,j)
positive variable q(i,j)

# assignment constraints
sum(i,p(i,j))=1
sum(j,p(i,j))=1

# quantities sold
x(j) = sum(i, q(i,j))
x(j) >= x(j+1)

# linearization of q(i,j) = p(i,j)*x(i)
q(i,j) <= p(i,j)*xup(i)
x(i) - xup(i)*(1-p(i,j)) <= q(i,j) <= x(i)

' interest rate
r(i) = 2*sum(j, p(i,j)*j)/100

Here xup(i) is an upper bound on x(i).

Not a very elegant formulation.