# Priority Constraint

Suppose I have the following set of binary variables:

$$X_i$$: $$I$$ ranges from {1,..,4} Highest priority among the three variables $$X$$ , $$Y$$ and $$Z$$

$$Y_j$$: $$J$$ ranges from {1,..,3}

$$Z_k$$: $$K$$ ranges from {1,2} lowest priority among the three variables $$X$$ , $$Y$$ and $$Z$$

How can I formulate the following :

(1) If any variable $$Z_k = 1$$ for each $$k\in K$$, Then each and every $$Y_j$$ variables $$y_1$$, $$y_2$$, $$y_3$$ must first $$=1$$

i.e. $$y_1 = 1$$, $$y_2 = 1$$, $$y_3 = 1$$

In other words, before any $$Z_k$$ for each $$k\in K$$ $$=$$ 1, all $$Y_j$$ variables must FIRST = 1

(2) SAME APPLIES FOR THE RELATIONSHIP BETWEEN $$X$$ AND $$Y$$ variables

If any variable $$Yj = 1$$ for each $$j\in J$$ Then each and every Xi variable $$X1$$, $$X2$$, $$X3$$, $$X4$$ must first $$=1$$

$$x1 = 1$$, $$x2 = 1$$, $$x3 = 1$$ , $$x4 = 1$$

In other words, before any $$Y_j$$ for each $$j\in J$$ variables = 1, all $$Yj$$ variables must first be = 1

I'll write an example just to make sure I was clear:

Before $$y_2$$ is picked and is = 1, All $$x_i$$ for each $$i\in I$$ must equal to 1. Meaning that X variables are at a higher priority than y variables and should be picked first.

• Welcome to OR.SE. Is there any difference between $x$ and $X$, and $y$ and $Y$? – A.Omidi Dec 1 '20 at 7:47
• No they are both the same, sorry for this mistake. – MAHER Dec 1 '20 at 22:59

When I get your problem correctly you want to enforce every $$X_i$$ to be activated when any $$Y_j$$ is activated and every $$Y_j$$ to be activated when any $$Z_k$$ is activated.
This can be achieved by adding constraints: \begin{align} X_i &\geq Y_j &\forall i\in I, j\in J\\ Y_j &\geq Z_k &\forall j\in J, k\in K\\ \end{align} Here the first set enforces all $$X_i=1$$ when any $$Y_j=1$$ and the second all $$Y_j=1$$ when any $$Z_k=1$$.