I want to use conditions to my variable.
dvar boolean x[I][J][K][L]
dvar in h[i]
my code is
forall(i in IP, j in J)
sum(k in K, l in L)
x[i][j][k][l] - 7 == h[i];
The following condition must be satisfied:
if x[i][j][k][l] - 7 <=0, h[i] must be zero.
if x[i][j][k][l] - 7 >=0, h[i] must be calculated
How do we formulate it in the integer programming formulation on CPLEX?
Thank you for your answer.
In OPL CPLEX you can use if then logical constraints.
For instance
range I=1..2;
range J=1..3;
range K=1..4;
range L=1..5;
dvar boolean x[I][J][K][L];
dvar int h[I];
subject to
{
forall(i in I, j in J,k in K,l in L)
{
(x[i][j][k][l] - 7 >=0)
=> (sum(k2 in K, l2 in L) (x[i][j][k2][l2] - 7) == h[i]);
(x[i][j][k][l] - 7 <=0)
=> (0 == h[i]);
}
}
works
It looks like you want to enforce $h_i = \max(\sum_{k,l} x_{i,j,k,l}-7, 0)$. Some solvers will automatically linearize this. Depending on where $h_i$ appears elsewhere in the problem, you might be able to get by with enforcing instead $h_i \ge \max(\sum_{k,l} x_{i,j,k,l}-7, 0),$ which you can do without introducing new variables, by imposing \begin{align} h_i &\ge \sum_{k,l} x_{i,j,k,l}-7 \tag1\\ h_i &\ge 0 \tag2 \end{align} If necessary, you can also enforce $h_i \le \max(\sum_{k,l} x_{i,j,k,l}-7, 0)$ by introducing binary variable $\delta_{i,j}$ (as in the proposed formulation by @anjikum) and imposing \begin{align} h_i &\le \sum_{k,l} x_{i,j,k,l} - 7\delta_{i,j} \tag3\\ h_i &\le (M_{i,j}-7)\delta_{i,j} \tag4 \end{align} Here, $M_{i,j}$ is a small constant upper bound on $\sum_{k,l} x_{i,j,k,l}$.
Below is one way to formulate it. Need to introduce another binary variable ($\delta_{i,j}$). $M$ depends on bounds of $h_{i}$ and $x_{i,j,k,l}$ : $$ \begin{align} \sum_{k,l} x_{i,j,k,l}-7 &\le h_{i} + M \cdot (1-\delta_{i,j})\\ \sum_{k,l} x_{i,j,k,l}-7 &\ge h_{i} - M \cdot (1-\delta_{i,j}) \\ \sum_{k,l} x_{i,j,k,l}-7 &\le M \cdot \delta_{i,j}\\ \sum_{k,l} x_{i,j,k,l}-7 &\ge -M \cdot (1-\delta_{i,j}) \end{align} $$ Regarding CPLEX code, you need to add these constraints similar to declaration in your code (provided in question).
