Using max/min in mixed integer programming model

Question

I building a mixed integer programming model, and I want to define the minimum and mzximum of a decision variable.

for example lets say the C={19, 20, 30}

I want to define C_early as 19 and C_late as 30. Then I want to minimize the difference. The C_late part was defined successfully using auxiliary constraints, however, I think I am missing something for the min part.

here is my code:

int I=...;
int J=...;
int K=...; 
int T=...;

range Order = 1..I;
range Job = 1..J;
range Machine=1..K;
range Position = 1..T;

int p[Order][Job]=...;
int a[Order][Job][Machine]=...;


dvar boolean x[Order][Job][Machine][Position];
dvar int C_late[Order];
dvar int C_early[Order];
dvar int diff[Order];
dvar int+ y [Machine][Position];
dvar int+ C[Order][Job];
dvar int Cmax;


minimize 
Cmax;

subject to{
// Ensure that a job is scheduled on one position only
forall(i in Order, j in Job: p[i][j]>0) sum(t in Position, m in Machine) 
x[i][j][m][t] == 1;

forall(m in Machine, t in Position) sum(i in Order, j in Job: p[i][j]>0)
x[i][j][m][t] <= 1;  

forall(i in Order, j in Job: p[i][j]>0 , m in Machine, t in Position)
   x[i][j][m][t] - a[i][j][m] <= 0;

forall (m in Machine)
y[m][1] >= sum(i in Order, j in Job) p[i][j]*x[i][j][m][1];

forall(m in Machine, t in Position: t>=2)
y[m][t] >= y[m][t-1] + sum(i in Order, j in Job) p[i][j]*x[i][j][m][t];

forall(i in Order, j in Job: p[i][j]>0, m in Machine, t in Position) 
C[i][j] >= y[m][t] - 100000*(1 - x[i][j][m][t]);

forall(m in Machine) 
sum(i in Order, j in Job, t in Position) p[i][j]*x[i][j][m][t] - Cmax <= 0;

forall(i in Order, j in Job: p[i][j]>0)
C[i][j] >= C_early[i];

forall(i in Order, j in Job: p[i][j]>0)
C[i][j] <= C_late[i];

forall (i in Order)
C_late[i] - C_early[i] <= diff[i]
}

Last three constraints are related to my question.

Example on dataset:

J=3;
K=3;
T=10;
I=10;

p= [
        [15,0,0], 
        [14,0,0],
        [16,0,0],
        [15,0,0],
        [14,0,0],
        [16,0,0],
        [16,0,0],
        [14,0,0],
        [15,0,0],
        [17,16,14]
            ];

a = [
        [[1,1,1], [0,0,0], [0,0,0]], 
        [[0,1,1], [0,0,0], [0,0,0]],
        [[1,1,1], [0,0,0], [0,0,0]],
        [[1,0,1], [0,0,0], [0,0,0]],
        [[0,1,1], [0,0,0], [0,0,0]],
        [[1,1,1], [0,0,0], [0,0,0]],
        [[0,1,1], [0,0,0], [0,0,0]],
        [[0,1,1], [0,0,0], [0,0,0]],
        [[1,1,1], [0,0,0], [0,0,0]],
        [[1,1,1], [1,1,1], [1,0,1]],
            ];

I know that I have to use big m method for the min constraint, however, I am not sure how Thanks,

What values are you getting for C_early and C_late for some example small problem instances? — TimChippingtonDerrick
– TimChippingtonDerrick, Commented Oct 7, 2015 at 13:40
I get the right values when I try to minimize Cmax, however, when I try to minimize this : (sum i in Order) diff[i] --> C_late and C_early will be equal. I think I need to apply big m method for the C_early constraint, however, I dont know how — H.D
– H.D, Commented Oct 8, 2015 at 22:41

JF Meier · Accepted Answer · 2015-10-07 17:05:35Z

0

You only minimize Cmax, not diff. This is probably an error.

answered Oct 7, 2015 at 17:05

JF Meier

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1 Comment

H.D Over a year ago

Cmax is an objective I used to test the model, I frogot to put the new one which is: (sum i in Order) diff[i]

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