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

Question

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

BlackTom AI · Accepted Answer

x1 must be an integer (0, 1, 2, ...), x2 must be nonnegative (real allowed), and x3 must be binary (0 or 1). Any candidate solution violating any of these three constraints is infeasible.

1) Relevant knowledge points and concepts
- Integrality constraint: x1 ≥ 0 and integer means x1 ∈ {0,1,2,...}. A value like 1.01 is not an integer and therefore violates this constraint.
- Nonnegativity: x2 ≥ 0 means any real number ≥ 0 is allowed (fractions acceptable).
- Binary constraint: x3 ∈ {0,1} (x3 = 0 or 1 only).

2) Step-by-step reasoning for each option
Check each solution against the three constraints in order (x1 integer & ≥0, x2 ≥0, x3 ∈ {0,1}):

Option A: x1 = 0, x2 = 1.60, x3 = 0
- x1 = 0: integer and ≥0 ✔
- x2 = 1.60: ≥0 ✔
- x3 = 0: binary ✔
→ Feasible

Option B: x1 = 2, x2 = 2.6667, x3 = 1
- x1 = 2: integer and ≥0 ✔
- x2 = 2.6667: ≥0 ✔
- x3 = 1: binary ✔
→ Feasible

Option C: x1 = 1.01, x2 = 3.3333, x3 = 0
- x1 = 1.01: not an integer ✖
- x2 = 3.3333: ≥0 ✔
- x3 = 0: binary ✔
→ Infeasible because of x1

Option D: x1 = 5, x2 = 0.1667, x3 = 0
- x1 = 5: integer and ≥0 ✔
- x2 = 0.1667: ≥0 ✔
- x3 = 0: binary ✔
→ Feasible

3) Calculations / checks shown
- For Option C: check integer condition: floor(1.01) = 1 and ceil(1.01) = 2, but 1.01 ≠ 1 and 1.01 ≠ 2, so 1.01 ∉ Z. Therefore x1 violates integrality.

4) Why the correct answer is correct
The only constraint violated across all options is integrality of x1 in option C. Since integrality is mandatory, any non-integer x1 makes the solution infeasible.

5) Why the other options are incorrect (i.e., why they are actually feasible)
Options A, B, and D all have x1 values that are integers ≥0, x2 values ≥0, and x3 equal to 0 or 1; therefore they satisfy all constraints and are feasible.

6) Conclusion
The solution that would NOT be feasible is x1 = 1.01, x2 = 3.3333, x3 = 0, which matches the provided answer.

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

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