As a project manager, Mary is managing 12 different projects. She realized something very interesting related to them. Sometimes they present some unique and particular rules, like the ones for projects #: 3, 6, and 8. She concluded that neither project 6 nor project 8 is selected unless project 3 is selected.

For integer programming problems, which one of the alternatives listed below correctly models this situation?

Question

As a project manager, Mary is managing 12 different projects. She realized something very interesting related to them. Sometimes they present some unique and particular rules, like the ones for projects #: 3, 6, and 8. She concluded that neither project 6 nor project 8 is selected unless project 3 is selected.

For integer programming problems, which one of the alternatives listed below correctly models this situation?

BlackTom AI · Accepted Answer

The implication that "neither project 6 nor project 8 is selected unless project 3 is selected" is modeled by logical implications x6 → x3 and x8 → x3. For binary decision variables (xi ∈ {0,1}), an implication p → q is equivalent to the linear inequality p ≤ q.

1) Relevant knowledge points and concepts
- Binary variables xi ∈ {0,1} represent selection (1 = selected, 0 = not selected).
- Logical implication p → q for binaries is (~p) OR q, which is equivalent to p ≤ q.
- Integer programming uses linear inequalities to encode logical rules.

2) Step-by-step reasoning
- The statement "project 6 is not selected unless project 3 is selected" means if x6 = 1 then x3 must equal 1; symbolically x6 → x3.
- For binary variables, x6 → x3 ⇔ x6 ≤ x3.
- Similarly, "project 8 is not selected unless project 3 is selected" gives x8 → x3 ⇔ x8 ≤ x3.
- Therefore the two constraints x8 ≤ x3 and x6 ≤ x3 correctly encode the rule.

3) Intermediate verification (truth cases)
- If x3 = 0, inequalities give x6 ≤ 0 and x8 ≤ 0 ⇒ x6 = 0, x8 = 0 (neither can be selected).
- If x3 = 1, inequalities give x6 ≤ 1 and x8 ≤ 1 ⇒ x6, x8 may be 0 or 1 (no restriction beyond binary).

4) Why the correct answer is correct
- The pair x8 ≤ x3, x6 ≤ x3 enforces exactly the intended implications and allows all valid combinations: when x3 = 0 both x6 and x8 must be 0; when x3 = 1 they can be chosen freely.

5) Why other options are incorrect
- x3 + x6 + x8 ≤ 1: forbids selecting more than one project; this is far stronger and wrong.
- x3 ≤ x8, x3 ≤ x6: forces x3 = 1 only if both x6 and x8 are 1 (opposite direction).
- x6 ≤ x8, x3 ≤ x8: links everything to x8, wrongly requiring x8 when x3 is chosen and tying x6 to x8.
- None of the above: incorrect because a correct representation is present.

6) Conclusion
The correct formulation is x8 ≤ x3 and x6 ≤ x3, which matches the provided answer.

类似问题

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.

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.

In a 0-1 integer programming model, which of the alternatives listed below correctly models the following statement? "There is no chance for x3 to be constructed unless you construct x5."

If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is not mutually exclusive constraint.

To perform sensitivity analysis involving an integer linear program, it is best to

In a model, x1 ≥ 0 and integer, x2 ≥ 0, and x3 = 0 or 1. Which of the following solutions would NOT be feasible?

Modelling a fixed cost problem as an integer linear program requires

If the profit of product Y goes up to $ 166, and you require integrality of X and Y, then what is the optimal profit?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单