Contingent Decisions¶

Definition¶

Contingent decisions are binary choices whose admissibility depends on other choices. If decision 2 is allowed only when decision 1 is yes, the implication is represented by

y_2\le y_1,\qquad y_1,y_2\in\{0,1\}.

Interpretation¶

Contingent decisions model the logical rule "decision 2 is only possible if decision 1 is taken", such as building a warehouse only where a factory is built. With binary variables for the two yes-or-no choices, the rule becomes the linear inequality $y_2\le y_1$ $y_{1}$ .

A 2 by 2 table of the values of two binary decisions y1 and y2, in which only the combination y1 = 0, y2 = 1 is forbidden by y2 at most y1.

The table lists all four value pairs of $(y_1,y_2)$ . Three of them satisfy $y_2\le y_1$ $y_{1}$ ; only $y_1=0$ $0$ , $y_2=1$ fails, since $1\le0$ is false. That is exactly the forbidden combination, decision 2 without decision 1. The pair $y_1=1$ $1$ , $y_2=0$ stays allowed, so the inequality does not force decision 2 once decision 1 is taken.

Checking every value combination, as in the table, is the general way to design and verify such constraints. In words, $y_2\le y_1$ says "if $y_2=1$ then $y_1=1$ $1$ ", or equivalently "if $y_1=0$ then $y_2=0$ $0$ ". If instead decision 2 must be taken whenever decision 1 is, the reverse inequality $y_1\le y_2$ is needed, and requiring both gives $y_1=y_2$ $y_{2}$ .

A common slip is to read $y_2\le y_1$ as a two-way link. It only rules out "2 without 1"; whether decision 2 is taken when decision 1 is depends on the objective and the other constraints. Rules with several conditions are built from several such inequalities, one per condition, as in the Key Results. Mutually exclusive decisions express a different rule, "at most one", with a sum. Both kinds of rule can be added to a project selection model when projects depend on or exclude each other.

Key Results¶

If decision 2 must be yes when decision 1 is no, use $y_1+y_2\ge1$ $1$ .
If decision 3 is allowed only when both decisions 1 and 2 are yes, use $y_3\le y_1$ and $y_3\le y_2$ $y_{2}$ .
If $y_1=1$ and $y_2=0$ must imply $y_3=0$ $0$ , use $y_1-y_2+y_3\le1$ $1$ .

Examples¶

With binaries $W_{LA}=1$ if a warehouse is built at location LA and $F_{LA}=1$ if a factory is built there, $W_{LA}\le F_{LA}$ requires a local factory before a warehouse can be built.
$y_1+y_2+y_3\le2$ rules out all three yes decisions together.

Prerequisites¶

Binary Variable

Sources¶

OR Modelling · Week 3

Symbols: element of

Last updated: 20 October 2026 03:02