Linear programming with minimum and maximum fixed cost - linear-programming

Question

I need your help in the following optimisation problem.
I have a maximisation Mixed Integer linear programming problem.
I would like to consider the minimum & maximum fixed fee.
I've done it in this way..
cost = max(minimum fixed cost , cost rate * x)
cost >= minimum fixed cost
cost >= cost rate * x
cost = min(maximum fixed cost , cost rate * x)
cost <= maximum fixed cost
cost <= cost rate * x
But, This turns infeasible solution.
Would you please help me in optimising such a problem.

Answer 1

piecewise linear functions
I think what you mean is the following:
Let
A = minimum fixed cost / cost rate
B = maximum fixed cost / cost rate
Then you want to model the piecewise linear function:
cost = minimum fixed cost if x < A
cost rate * x if A <= x <= B
maximum fixed cost if x > B
Using piecewise linear functions inside a MIP model is not a problem. You can do this by different approaches:
using extra binary variables (see (1))
using SOS2 variables (see (1))
systems like AMPL and Gurobi have special facilities to express piecewise linear functions.
Example formulation
A formulation with SOS2 variables can look as follows:
Introduce data points px and py
px py
--------------------------
0 minimum fixed cost
A minimum fixed cost
B maximum fixed cost
C maximum fixed cost
where we assume 0<=x<=C. I.e. C is an upper bound on x.
Then do:
set p = {1,2,3,4}
sos2 variables lambda(p)
sum(p, lambda(p)) = 1
x = sum(p, lambda(p)*px(p))
cost = sum(p, lambda(p)*py(p))
See e.g. (2)
What is wrong with your approach
Note that your approach (shown in the question) is incorrect:
cost >= minimum fixed cost
cost >= cost rate * x
cost <= maximum fixed cost
cost <= cost rate * x
is really
minimum fixed cost <= cost <= maximum fixed cost
cost = cost rate * x
which limits x to A <= x <= B.
References
(1) H.Paul Williams, "Model Building in Mathematical Programming", Wiley
(2) GAMS: Piecewise linear functions with SOS2 variables

Answer 2

Piecewise linear function:
cost = 0.02 x if 0 <= x <= 500
0.03 x if 500 <= x <= 1500
0.04 x if 1500<= x <= 10000
SOS2 Solution:
x[a, b, c, d, e] = (x1[a, b, c, d, e] * 0) +
(x2[a, b, c, d, e] * 500) +
(x3[a, b, c, d, e] * 1500) +
(x4[a, b, c, d, e] * 10000)
cost[a, b, c, d, e] = (x1[a, b, c, d, e] * 0 * 0 )+
(x2[a, b, c, d, e] * 500 * 0.02)+
(x3[a, b, c, d, e] * 1500 * 0.03)+
(x4[a, b, c, d, e] * 10000 * 0.04)
x1[a, b, c, d, e] + x2[a, b, c, d, e] + x3[a, b, c, d, e] + x4[a, b, c, d, e]>= 0
x1[a, b, c, d, e] <= y1[a, b, c, d, e]
x2[a, b, c, d, e] <= y1[a, b, c, d, e] + y2[a, b, c, d, e]
x3[a, b, c, d, e] <= y2[a, b, c, d, e] + y3[a, b, c, d, e]
x4[a, b, c, d, e] <= y3[a, b, c, d, e]
y1[a, b, c, d, e] + y2[a, b, c, d, e] + y3[a, b, c, d, e] = 1
I've done it in this way. However, the solver still shows unfeasible solution. Can you see something wrong in this formulation?!