In a 0-1 integer programming model, if the constraint x2 + x3

Question

In a 0-1 integer programming model, if the constraint x2 + x3 <= 1.

It means that if project 2 is not selected, project 3 ______________.

Choose the option below that best represents the empty space above.

BlackTom AI · Accepted Answer

Binary variables x2 and x3 can each be 0 or 1. The constraint x2 + x3 <= 1 means at most one of the two projects may be selected (they cannot both be 1).

1) Relevant concepts
- 0-1 integer (binary) variables: each xi ∈ {0,1}.
- Constraint x2 + x3 <= 1 enforces mutual exclusivity: the sum cannot exceed 1, so you cannot have both selected simultaneously.

2) Step-by-step reasoning
- Assume “project 2 is not selected” → x2 = 0.
- Substitute into the constraint: 0 + x3 <= 1, which simplifies to x3 <= 1.
- Because x3 is binary, the feasible values satisfying x3 <= 1 are x3 = 0 or x3 = 1.

3) Intermediate steps shown
- Start: x2 + x3 <= 1
- With x2 = 0: 0 + x3 <= 1
- Therefore: x3 <= 1
- Given x3 ∈ {0,1} → x3 ∈ {0,1} (both allowed)

4) Why the correct answer is correct
The constraint does not force x3 to a single value when x2 = 0; it merely allows x3 to be either 0 or 1. In plain language: if project 2 is not selected, project 3 may or may not be selected. That exact choice is not listed among the options, so none of the listed specific statements correctly describe the implication.

5) Why the other options are incorrect
- “will not be selected”: implies x3 must be 0 when x2 = 0 — false.
- “will be selected” / “will always be selected”: imply x3 must be 1 — false.
- “is more likely to be selected”: introduces a probabilistic notion not implied by a deterministic constraint; the constraint gives no probability or preference.

6) Conclusion
Because the constraint allows x3 to be either 0 or 1 when x2 = 0 and none of the provided specific claims matches that, the correct choice is: None of the above. This matches the provided answer.

In a 0-1 integer programming model, if the constraint x2 + x3 <= 1. It means that if project 2 is not selected, project 3 ______________. Choose the option below that best represents the empty space above. Single choice

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