Priority Constraint

Question

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.

Answer 1

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 $Z_k=1$.