Let xi = 1 if Project i is selected, i = 1,2,3,4,5; and 0 otherwise:

Which answer below indicates that if Project 4 is selected, then Project 2 must also be selected?

Question

Let xi = 1 if Project i is selected, i = 1,2,3,4,5; and 0 otherwise:

Which answer below indicates that if Project 4 is selected, then Project 2 must also be selected?

BlackTom AI · Accepted Answer

Implication x4 = 1 ⇒ x2 = 1 is represented by the linear inequality x4 ≤ x2.

Relevant knowledge points:
- Binary decision variables xi take values 0 or 1.
- A logical implication "if A then B" (A ⇒ B) for binary variables is equivalent to the inequality A ≤ B (or B − A ≥ 0).
- In integer/linear programming, logical conditions must be written as linear constraints.

Step-by-step reasoning:
1. Interpret the logical statement: "If Project 4 is selected (x4 = 1), then Project 2 must also be selected (x2 = 1)." This is an implication: x4 = 1 implies x2 = 1.
2. For binary variables, list possible (x2, x4) pairs and which should be allowed:
   - (0,0): x4 = 0 so implication places no requirement — allowed.
   - (1,0): x4 = 0 — allowed.
   - (1,1): x4 = 1 and x2 = 1 — satisfies implication — allowed.
   - (0,1): x4 = 1 but x2 = 0 — violates implication — disallowed.
3. Find an inequality excluding only the disallowed pair (0,1). The inequality x4 ≤ x2 yields:
   - (0,0): 0 ≤ 0 true
   - (1,0): 0 ≤ 1 true
   - (1,1): 1 ≤ 1 true
   - (0,1): 1 ≤ 0 false
   So x4 ≤ x2 exactly enforces the required implication.
4. Algebraically, x4 ≤ x2 can be written as x2 − x4 ≥ 0, a linear constraint suitable for integer programming. It enforces x2 = 1 whenever x4 = 1, but allows x2 = 1 without x4.

Why the correct answer is correct:
- x4 ≤ x2 forbids the only illegal case (x4 = 1 and x2 = 0) and permits all legal assignments, matching the desired logical implication.

Why the other options are incorrect:
- x2 + x4 = 1 forces exactly one of the two projects selected, which contradicts the possibility that both may be selected.
- x2 + x4 ≤ 1 forces at most one selected, again disallowing the valid case x2 = x4 = 1.
- x2 ≤ x4 is the reverse implication (x2 = 1 ⇒ x4 = 1), not the required direction.
- "None of the above" is false because x4 ≤ x2 correctly models the implication.

Conclusion: The correct constraint is x4 ≤ x2, which matches the provided answer.

Let xi = 1 if Project i is selected, i = 1,2,3,4,5; and 0 otherwise: Which answer below indicates that if Project 4 is selected, then Project 2 must also be selected?单项选择题

类似问题

Let x1 and x2 be 0-1 variables whose values indicate whether projects 1 and 2 are not/are completed. Which of the following answers indicates that project 2 can be completed only if project 1 has been completed?

Binary variables:

Binary variables:

Assuming W1, W2, and W3 are 0-1 integer variables, we have a constraint W1 + W2 + W3 ≤ 2 in the model. Which of the following statements is correct?

Muscles innervated by the lateral branches of the dorsal/posterior rami include:

Which of the following is the odd one out in terms of motor innervation by ventral/anterior rami?

Which cranial nerve innervates the geniohyoid muscle?

Internal iliac arteries branch from:

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