For the given 0-1 integer programming, f the constraint x1 - x2 ≤ 0, it means if project 1 is selected, project 2 ________________.

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

Question

For the given 0-1 integer programming, f the constraint x1 - x2 ≤ 0, it means if project 1 is selected, project 2 ________________.

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

BlackTom AI · Accepted Answer

x1 - x2 ≤ 0 can be rewritten as x1 ≤ x2. For 0–1 (binary) variables this relationship forces x2 to be at least as large as x1.

1) Relevant knowledge points and concepts
- In 0–1 integer programming, each variable xi takes value 0 (not selected) or 1 (selected).
- An inequality relating two binary variables restricts which combinations of 0/1 are feasible.
- Algebraic manipulation: x1 - x2 ≤ 0 is equivalent to x1 ≤ x2.

2) Step-by-step reasoning
- Start from the constraint: x1 - x2 ≤ 0.
- Add x2 to both sides: x1 ≤ x2.
- Because x1 and x2 are binary, the only possible values are 0 or 1. If x1 = 1, the inequality becomes 1 ≤ x2, so x2 must equal 1. If x1 = 0, x2 can be 0 or 1.

3) Intermediate calculations / enumeration of cases
- (x1, x2) = (0,0): 0 - 0 = 0 ≤ 0 → feasible.
- (x1, x2) = (0,1): 0 - 1 = -1 ≤ 0 → feasible.
- (x1, x2) = (1,0): 1 - 0 = 1 ≤ 0 → infeasible.
- (x1, x2) = (1,1): 1 - 1 = 0 ≤ 0 → feasible.
Thus the only disallowed case is selecting project 1 while not selecting project 2.

4) Why the correct answer is correct
Because x1 ≤ x2 and x2 is binary, whenever x1 = 1 you must have x2 = 1. In plain language: if project 1 is selected, project 2 must be selected.

5) Why the other options are incorrect
- "must not be selected": contradicts x1 ≤ x2; that would require x2 = 0 when x1 = 1, which is infeasible.
- "can sometimes be selected": incorrect because selection of project 1 forces project 2; there is no “sometimes” when x1 = 1.
- "is unlikely that it will be selected": subjective and irrelevant to a hard constraint.
- "None of the above": incorrect because a correct option is present.

Conclusion: The constraint implies project 2 must be selected, matching the provided answer "must be selected."

For the given 0-1 integer programming, f the constraint x1 - x2 ≤ 0, it means if project 1 is selected, project 2 ________________. Choose the option below that best represents the empty space above.单项选择题

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